a new model for laminar, transitional, and turbulent flow of drilling muds

8/10/2019 A New Model for Laminar, Transitional, And Turbulent Flow of Drilling Muds

1/14

Society of Petroleum Engineers

SPE 25456

A New Model for Laminar Transitional and Turbulent Flow

of Drill ing Muds

T.O.

Reed, Conoco Inc.

and

Pilehvari U. of Tulsa

SPE

Members

Copyright

1993,

Society of Pf1.\roleum Engineers Inc.

This peper

was

prepared for presentation

at

the Production Operations Symposium held

In

Oklahoma City

OK,

U.S.A. March 21 23

1993.

This paper

was

selected for presentation

by

an

SPE Program

Commillee following review of information contained In an abstract submitted

by

the

author s .

Contents

of

the paper

as presented

have

not

been reviewed

by the Society of Petroleum Engineers

and

are SUbject to correction

by

the

author s . The

material as presented does not necessarily reflect

any p o s ~ l o n of the Society of Petroleum Engineers Its officers ormembers. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society

of Petroleum Engineers. Permission to copy

Is

restricted to an abstract of notmore than 300words. illustrations may notbe copied. The ebstrect shouldcontain conspicuous acknowledgment

of and by whom the paper is presented. Write Librarian SPE, P.O. Box 833836, Richardson TX 75083-3838, U.S.A. Telex 163245 SPEUT.

STR CT

The concept of an Effective diameter

is

introduced for the flow of

drilling muds through annuli.

This

new

diameter

accounts for both

annular geometry and the effects of a non-Newtonian fluid. It

provides the link between Newtonian pipe flow and non-Newtonian

flow through concentric annuli. The method

is

valid in any flow

regime and can

be

used to determine whether a DOn-Newtonian flow

is

laminar, transitional, or turbulent. n analytical procedure is

developed for computing frictional pressuregradients in all

three

flow

regimes.

The

analysis

also

quantifies bow

flow

transition

is

delayed

by

increasing the yield stress

of

a fluid. In

addition,

it

is

shown that

transition in an annulus is delayed to higher pump rates as the ratio

of

inner to outer diameter increases. Furthermore, the method

accounts for wall roughness and its affects on transition81

and

turbulent pressure gradients for non-Newtonian

flow

through pipes

a pipe

or

concentric annulus

is

laminar, transitional, or fully turbu

lent.

A derivation

of

the model is presented in the Appendix. Some

additional background and results from the model are given in the

fonowing discussions.

NEWTONIAN FLOW

RelatioDShip Between

Pipe

and

nnular

Flows.

A

considerable

number

of

equivalent diameters

have

beenproposed over theyears

for

flow through conduits other than circular

pipes.1O-1z

The purpose

of

defining such a diameter

is to

introduce definitions of friction

factor and Reynolds number thatwill enable application of thewen

known relations

for

pipe

flow

to other geometries. In particular, the


2/14

2

A NEW

MODEL

FOR

LAMINAR,

TRANSmONAL, AND

ruRBULENT FLOW

OF

DRILLINGMUDS SPE2S456

developed by Hanks in th e early 1960s.

7,1 17

The solid curve comes

from simply setting

the

Reynolds number,

based

on

th e

Equivalent

Diameter of Jones

and

Leung, equal

to

2100. A Reynolds number

based

only on Lamb's Diameter is

used

for the ordinate in order

to

display how the critical values vary with diameter ratio. I t is readily

seen that the much simpler transition criterion provides better

agreement withHanks experimental data.

The

new criterion shows

that transition from laminar flow is delayed

to

progressively higher

velocities or pump rates as diameter ratio increases.

NON-NEWTONIAN

FLOW

Pipe

Flow.

Prof. Metzner

and

his students reported

some ~ i o n r i n

work in

the

19508

on

non-Newtonian flow through pipes. l-Z1 Their

experiments with Power-Law fluids in pipes provided the first clear

definitions of how the friction factor varies with Reynolds number in

the laminar, transitional, and turbulent regimes. They found that a

decreasing power-law exponent delays transition to higher Reynolds

numbers and shifts

the

turbulent friction factors downward.

This

is

a direct result of increasing degrees of the phenomenon called wshear

thinning.

In addition to the test data,Metzner and his students also developed

a novel analysis that provided a

way

to generalize their results for

Power-Law fluids

to

all

time-independent, non-Newtonian fluids.

They simply defined:

Wall Shear Stress

=

K x (Newtonian Shear Rate)N (1a)

or using standard symbols

, N

T

w

= K x 8 v / D 1b)

From this equation, it follows that the definition of the exponent

WNW

is:

N

=

d(ln Twl / d(ln (8 v / D)] (1e)

They showed how the two parameters

K'w

and WNW can

be

applied to

Power-Law PL) fluids and Bingham Plastics (BP). The Appendix

physical diameter for di1atant fluids N > 1). As will be shown, the

Effective

Diameter

makes

i t easier

to

relate non-Newtonian and

Newtonian flows.

The

advantage of such a connect ion is tha t the

well-established friction factor relationsfor Newtonian flows can then

be

applied to non-Newtonian flows.

This

greatly simplifies the

task

of computing frictional pressure drops for such flows.

The

remaining variable that is needed to define th e Generalize

Reynolds

Number

is the apparent Newtonian viscosity. This

parameter is defined

by:

App. Viscosity =Shear Stress/Shear Rate @ Wall (4a)

1J.w,app

=

K

(8

v /

D)N /

Yw

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

u

(4b)

With these definitions, the Generalized REynolds number

GRE)

is

defined as:

NRe,O = v Deffl

1J.w,app

(5)

This expresses the Reynolds number for non-Newtonian pipeflow in

the

same algebraicfo rm as for a Newtonian fluid Metzner & Reed g

showed t ha t the GRE is related to laminar friction factor by the

same

classical equation for Newtonian flow,

viz,

c

=16/

NRe,o

, Laminar Non-Newtonian

Flow......

(6)

These

authors showed that experimental pipe flow

data

for a variety

of non-Newtonian fluids followed this relationship. Later experi.

ments with Power-Law fluids in turbulent pipe flow led Dodge

Metzner3l to modify the classical Colebrook equation for turbulent

friction factors in Newtonian pipe flow, see Eq.

A 45

This

was

necessary in order to correlate the data because the shear-rate exp0-

nentwas

found to

be

a significant parameter.

In

particular, friction

factors decreasewith decreasingvalues of the Power-Law exponent.

Figure

Z

shows

the

correlations

Dodge

Metzner developed for

laminar and turbulent flow.

The

transition zoneswere no t correlated

by these authors,

but

their experimental data follows the indicated

curves between the

end of

laminar flow and th e beginning of

fully-

turbulent low. As described in the Appendix, a new correlation for

the transition zones bas

been

developed

based

on th e experimental


3/14

SPE 25456

T. REED AND PILEHVARI

3

of

the Fanning friction factor and the

GRE

equals 16.1.

(As

may be

seen from Eq. 6, this product is 16 in Jaminar flow, and it exceeds this

value wben transiqon begins.)

This

generalized transition criterion is

designed

to

reduce

to

a critical Reynolds number of 2100 for

Newtonian pipe flow, see last section of Appendix.

The underprediction

of

the data by the Hanks and

Pratt

criterion

is

even

more

apparent when critical values of the GRE are plotted as

a function.of Hedstrom number,

Fig.

4. Here it becomes clear that

tbe

Hanks

and Pra tt criterion predicts decreasing

flow

rates

for

transition as the Yield Point (YP) increases. This certainly violates

tbe expected trend. The new generalized criterion show the expected

trend, i.e., higher pump rates, are required

to

achieve transition as

tbe

yP

of

a

BP

increases.

It

is

instructive to plot the friction factors for BP pipe flow through

all

three

flow regimes with

Hedstrom

number

as

a parameter . This

is sbownin

Fig.

5. This figure shows bow

an

increasing

yP

delays

transition to higher

pump

rates and

also

extends

tbe

transition zone

over a progressively wider range

of

pump rates. Fo r example, wben

tbe Hedstrom number is 500,000, the flaw is no t fully turbulent until

a Reynolds

number

of 100,OOO This compareswitha commonvalue

of about 3,000 for Newtonian pipe flow.

It is

particularly important

to

observe that all

of

the curves for

~ n s t o n

eventually merge into

tbe

turbulent curve for Newtonian

flow. This occurs because

the

y P becomes progressively less

significant as sbear rate increases. Hence, the curves have the

expected asymptotes. . There is one other point

to

notice from

comparing Figs. 5 and 3.

A

BP bas a sbear-rate exponent

of 1;

wbereas, a PL fluid generally bas a nonunity exponent. This is

the

basic reason wby the friction factor curves for different values of the

sbear-rate exponent do no t merge and remain distinct

for

all

Reynolds numbers.

In tum,

this implies that a Herschel.Bulkley fluid

will exhibit somewbat different behaviorsince it combines

tbe

BP and

tbe PL rbeological models.

In

particular, one can infer that a yield

stress

will

delay and extend

the

transition zone, and tb e transitional

values of friction factor will eventuallymerge with the fully-turbulent

curve for a PL fluid with an exponent equal

to

whatever appears in

tbe Herscbel-Bulkley model,

see

Eq. 7 below.

At tbis point, it is appropriate

to

introduce some viscometer data for

0566.

The results for the BP model is as expected i.e., transition is

delayed

tbe

most

by

using this model.

Furthermore, the

extent of .

the transition zone is expanded significantly, and the GRE is an

order-of-magnitude larger before fully-turbulent flow is achieved.

In

order

to

relate these results

to

a typical field application, consider

flow through a standard weight 5-incb drill pipe with an ID

of

4.276 incbes. Fu1Iy-turbuient flow is achieved at a GRE of3800 and

a pump rate Of 270

gpm

[17.0 lis] using

the

PL model.

The

corre

sponding results for

the

HB model occurs at a

GRE

of 4100 and a

pump

rate of 310 gpm [19.6 lis].

In

contrast, tbe BP model leads

to

a prediction

of

fully-turbulent flow

at

a

GRE

of 22,300 and a

pump rate of 670

gpm

[42.3

Urn].

(Note:

The

analysis does not

account for

the

effects

of

restrictions

at

tool jointswhicb could cause

flow transition

to

begin at lower pump rates.)

I t is

also

of interest to compare

the

frictional pressure gradients for

a realistic pump rate of

600

gpm [37.85 lis]. The results for

the

PI..,

HB, and BP models, respectively, are 0.085, 0.099, and 0.104 psilft

[22.62 kPa/m]. The relative close values for the HB and

BP

fluids

are

a result

of the

HB being fully turbulent and

the BP

being in

transition at this pump rate. Again

if

tool joints caused transition

to

occur at lower

pump

rates and the BP were in fully-turbulent flow at

600

gpm

[37.85 lis],

then

the difference between pressure drops for

the HB and BP would

be

larger.

Annular Flow. The flow model can

be

extended to concentric annuli

by altering tbe definition of the Effective d iameter

to

include the

effects of the different geometry. The Effective diameter for a

concentric annuli is equal

to

DJIy

IG.

The

Effective

diluneter for

non ewtonian flow

through

a concentric annulus is the dia

meter a

circular

pipe that would have the identical

pressure

drop forflow a ewtonian fluid with a viscosity equal to the

effective viscosity

which

is based on the average wall shear

rate in t annulus and has a

velocity

equal to the non-

ewtonian annular flowvelocity.

The

an

function

is

based on a correlation of the analytical solution

by

Fredrickson

and

Bird for flow

of

a Power-Law fluid through a

concentric annulus. This function is dependent

on both tbe

ratio of

inner to outer diameters and the shear-rate exponent.

The

function

is defined in

the

Appendix by

Eq.

A-34.

This correlation can be


4/14

4

A NEWMODELFOR LAMINAR, lRANsmONAL, AND

TURBULENTFLOWOF DRTI..LING

MUDS

SPE2S456

transition.

is

reasonable to expect

the

energy added to the flow by

the agitation may cause transition at

lower

pump rates.

Another parameter in the model is

the

ratio of Effective diameter

over

theEquivalent diameter. This ratio

partia1Iy

removes geometric

effects and

highlights

the non-Newtonian effects.

ip reshows

this

ratio approaches unity

as

N goes to one. This is the expected

limiting

condition for Newtonian

flow.

Since Fig. 8

shows

that N

increases

with

pump rate, the Effective diameter also increases

with

pump rate. This

has

the net effect of reducing the magnitude of

the

increase

in

frictional pressuregradient as pump rate

is

increased,

i.e.,

the fluid is shear thinning.

This

leads to the question, what happens to the apparent

viscosity

as

pump rate increases?

An Effective viscosity is

defined

in

a

man

ner analogous to pipe

flow.

The primary difference

for

an annulus

is:

the Effective

viscosity is

defined by the average

wall

shear stress

acting on the inner and outer walls and divided by

the

average wall

shear rate, see Eq. A-43. This viscosity is presented in

Fig

10 as a

function of pump rate. continuously decreaseswith pump rateand

.has a noticeable drop at transition

from laminar

to turbulent

flow.

This is caused by the rapidly

increasing

shear rate as transition

occurs.

Figure 11 presents friction factors

as

a

function

o the GRB

for

the

example

annulus and mud properties. For smoothwalls, the increase

in

friction

factor through the transition zone

is

small.

s

indicated

in

Fig. 8, the transition zone occurs between pump rates of 1910 and

2010 gpm [120.5-126.8 I.Js].

An example of the effects of roughwalls

is also

included

in

Fig.

11.

t may be noticed that the initiation of transition occurs at the same

GRB

This is

consistent

with

the experimental data of Nikuradse for

Newtonian pipe flow, see Schlichting.

24

sindicated in the figure,

transition to turbulent

flow

takes place over a smaller range of pump

rates,

1910

to

1970 gpm [120.5

to

124.3 I.Js],

and the

fully

turbulent

pressure drops are approximately

70

percent larger than the corre

sponding

turbulent

friction

factors for smooth

walls.

The

relative

roughness for

this

illustration

is approximately

0.01.

This is based on dividing the absolute roughness height by the

Equivalent

diameter, Eq.

A-14.

However, the correct definition of

relative roughness for non-Newtonian

flows

is the ratio of absolute

model

predicts

transition at higher pump rates and

friction

factors

that continuously

decrease

through the transition

zone

According

to

the

model,

the flow is not fully turbulent until a GRB of 18,000

and a pump rate

of

77JJ

gpm [45.42 I.Js].

The predicted

values

of

turbulent friction factors are

nearly

constant

for

a wall

roughness

of

0.00018 in [0.0046

mm].

The

above

value ofwall roughness gave good agreement between

predicted turbulent friction

factors

and experimentaldata for the

low-

rheology fluid. Unfortunately,

water

tests to determine wall

roughness for the annulus were not performed. However,

Fig.

12

shows encouraging agreement between predictions for the

low

rheology

fluid

and the corresponding data through u three flow

regimes.

Additional testswere conductedwith bentonitemuds

flowing

through

rough pipes. n these

cases wall

roughnesswas determinedvia water

tests, and predictions from the model,

using

the experimental

values

of wall roughness, agree

v ry

closely with the test data

for

u three

flow

regimes. This data covers a GRB range of 500 to 270,000 and

includes relative roughness values up to 0.002. Because of space

limitations, these non-Newtonian pipe-flow data are not presented

here.

CONCLUSIONS

1.

A new analysis

has

been developed

for

non-Newtonian flow

through pipes and concentric annuli. The method is based oil

relating non-Newtonian

flows to

Newtonian

flows. The advan

tage is that well-established results or Newtonian flows can be

applied to non-Newtonian

flows.

2 The Effective diameter is a key concept of the

method. t

accounts for both geometric and non-Newtonian fluid

effects

on frictional

pressure gradients in pipes and

annuli.

3.

Results agree

with

finite-difference computations for

laminar

flow of a Herscbel-Bullcley fluid through

concentric

annuli.

4. The

analysis

is valid for the laminar, transitional, and fully

turbulent

flow

regimes. Themethod incorporatesa

new transi-

tion

criteria that accounts for a delay in

flow

transition with


5/14

SPE 25456 T REED ND A PILEHVRI 5

7

w

= shear stress at walls of a pipe or annulus

11. The method runs on a 386 PC in only a.few seconds.

This

is

made

possible

by

avoiding either

f i n i t ~ r n

or

finite

element numerical solutions. Furthermore, the capabilities of

these numerical techniques ha\ie not yet been developed to

model transitional

nor

turbulent flow of a Herschel-Bulkley

fluid.

NOMENCLATURE

JLw.eff

p

7

= effective Newtonian viscosity

at

walls of an annulus,

Eq.

A-43

= fluid density, Ibm/gal

= shear

stress, Ibf/l00

tt2

A

=

parameter

used

to define N for a Herschel-Bulkley

fluid,

Eq.

A-31

D

=

internal diameter

of

a circular pipe, in.

D

eff

=

Effective diameter

of

either a pipe

or

annulus,Eq. A-37,

in.

Dhy

=

hydraulic diameter,

Do

- D

j

in.

D

j

=

inner diameter of annulus = drill pipe OD in

D

L

=

Lamb s diameter for a concentric annulus, Eq. A-l2 in

Do

=

outer diameter of annulus = borehole diameter

F

c

=

Fanning friction factor,

7..... / 1/2

p

v2

Flam

=

Fanning friction factor in laminar flow

Fir

=

Fanning friction factor in transition reginie

F

lUrb

=

Fanning friction factor in turbulent flow

G

=

an ExIog correlation factor, Eq. A-34

He

=

Hedstrom number, Eq. A-51

K

=

consistency index

used

foreithera Power-Law,Eq. A-19,

or

a Herschel-Bulkley fluid, Eq. A-29, Ibf-sec /l00 ft2

K

=

generalized

o n s i s t n ~

index of Metzner and

Reed

Eq. Ib Ibf-sec /l00

ft

L

=

length of pipe

or

annulus over which pressure

drop is

measured

m

=

flow behavior index for Herschel-Bulkley fluids, Eqs. 7

andA-29

ACKNOWLEDGEMENTS

We

thank

themanagement of Conoco, Inc. for permission to publish

this paper. We also thank Amoco, Inc. for allowing tests in their

research facilities

by

R

Subramanian, a

Tulsa

University graduate

student

REFEREN ES

1. Fredrickson, A G., and Bird, R B., Non-Newtonian Flow in

Annuli, IntI. Engr Chem. (March 1958) SO, No.3 347-52.

2. Hanks R W., The Laminar-Turbulent Transition for Fluids

with a Yield Stress, AlChE

1

May 1963) 9, No.3 306-09.

3.

Hanks,

R W. and Pratt, D. R On the Flow

of

Bingham

Plastic Slurries in

Pipes

and

Between Parallel Plates,

SPFJ

(Dec. 1967) 342-46.

4. Hanks,R W. and Ricks, B.L Laminar-Turbulent Transition

in Flow of Pseudoplastic Fluids with Yield Stresses, AlAA

Jour. ofHydronaulics (Oct. 1974) 8, No.4 163-66.

5. Hanks,

R

W., TheAxialLaminarFlow

of

Yield-Pseudoplastic

Fluids in a ConcentricAnnulus, IntI.

Engr

CMm

Process

Des.

Dev (1979) 18,

No.3

488-93.

6. Haciislamoglu,M.,

and

Langlinais, J., Non-NewtonianFlow in

Eccentric Annuli, ASME J. Energy Resources Tech., Sept

1990)

lU No.3

163-169.

7. Oltafor,M. N. and Evers, J. F., Experimental Comparison of

Rheology Models for Drilling Fluids, paper SPE 24086

presented at the 1992Western Regional

Meeting

BakerSfield,

CA, Mar. 3O-Apr. 1.


6/14

6

A NEW MODEL

FOR

LAMINAR,

TRANSmONAL

AND

TIJRBULENT FLOWOF

DRILIJNG

MUDS

SPE

25456

15.

Hanks, R. W. and Peterson, J. M., Complex Transitional

Flows in ConcentricAnnuli,

lChE

1.

(Sept. 1982) 28

No.5

SOO-06.

16.

Hanks, R.

W.

Tbe Laminar-Turbulent Transition

in

Noniso

thermal Flow of Pseudoplastic Fluids in Tubes, lChE

1.

(Sept. 1962)

8,

No.4 467-71.

17. Hanks, R.

W.

TbeLaminar-Turbulent Transition for Flow

in

Pipes, Concentric Annuli, and Parallel Plates,

lChE

lour

(Jan. 1963) 9, No.1 45-8.

18.

Hanks, R. W. ATheory of Laminar Flow

Stability

lChE

1.

(Jan. 1969)

15,

No.

I

25-8.

19.

Metzner,

A.

B., and Reed, J.

C.,

Flow

of

Non-Newtonian

Fluids - Correlation of the Laminar, Transition and Turbulent

flow Regions,

lChE

1. (Dec. 1955) I No.4 434-40.

20. Metzner, A. B., Non-Newtonian Fluid

Flow

[nil. Eng.

Chem

(Sept. 1957)

49, No.9

1429-32.

21.

Dodge, D. W. and Metzner,

A.

B., Turbulent Flow

of

Non

Newtonian Systems lChEl (June 1959) 5, No. 2 189-204.

22.

Mishra, P., and Tripathi, G., Transition from Laminar to

Turbulent Flow of Purely

VISCOUS

Non-Newtonian Fluids

in

Tubes, Chem Engt . Sci. (1971) ZCi, 915-21.

23.

Franco, V., and Verduzco, M. B., Transition Critical Velocity

in Pipes Transporting Slurrieswith Non-Newtonian Behavior,

Proc. 14th Miami Univ., Coral Gables, Multipbase Transport

.Particulate Phenomena Int l Symp. Vol. 4,

289-99,

Miami

Beach, FL, 1988.

24. Schlichting, H., BoundtllY Layer Theory, 7th eel McGraw-Hill

.Book

Co.

New York (1979), 616-20.

25.

Bourgoyne, A. T. Jr., Chenevert, M. R Millheim,

K. K.

and

Young, F. S. Jr.,

Applied Drilling Engineering,

Society

of

Pe.troleum Engineers, Richardson, IX p.

140 1986.

26.

Govier,

G.

W. and

Aziz,

K.

TheFlow

of

Complex

MIXtures

in

Pipes Van Nostrand Reinhold Co., New York (1972) 201-2.

27.

Hanks, R. W.

Low

Reynolds Number Turbulent Pipeline

Flow of Pseudohomogeneous Slurries,

Proc.

Hydrotransport

5

Fifth

Int l Cont on the Hydraulic Transport of Solids in

tive diameter

which properly accounts for non-Newtonian effects.

The developmentof the procedure beginswith

an

analysiS of

Iaminar

flow

in pipes and annuli. Next, fully-turbulent flow

is

analyzed and

wall

roughness is introduced. FJnally a procedure is developed for

predictingwhen transition from laminar to turbulent flow beginsand

when it ends.

LAMINAR NEWrONIAN FLOWS

Pipe Flow. The purpose of the following discussion is to introduce

the terminology that will

be

used. The solution forisothermaI

fully

developed viscous

flow of

a Newtonian

fluid

through a

circUlar

pipe

was derived

by

Hagen and Poiseuille in the 1800s. Their equation

for frictional pressure gradient

is:

dP /

elL

=32,.,. v

2

(A-I)

Next, we note that a Newtonian fluid

is

defined to be one for

which

shear stress

is

linearly proportional to shear rate,

i.e.

Shear Stress =

viscosity x

Shear rate.

T

=

. . (A-2)

Equations A-I and A-2 can be related

via

a simple force balance

between the pressure gradient that

drives

flow

through the pipe and

the viscous

forces

at the

wall

that oppose themotion.

This

provides

the

following

equation.

Tw =dP / elL X 0 /4 A-3

The subscript OW refers to the shear stress and shear rate at the

inside

wall

of a pipe. An expression for shear rate

at

the

waU

can be

obtained

by

combining Eqs. A1, A-2, and A-3, viz.

w

=

8

v /

0 (A.4)

The next step

is

to introduce the Fanning friction factor,

which is

defined

byEq. A-5.

F

f

= T

w

/ Ia p v2 =dP / elL X 0 / (2 P v2

A-S


7/14

SPE 25456

T. REED AND A PILEHVARI

7

As

before, a force balance between the pressure gradient and the

opposing viscous shear stresses at the innerand outer boundaries of

an annulus can be constructed

to

obtain a relation between them.

The results are:

Tw,avg

= To Do + Tj D

j

I

Do

+ D

j

=

dP

I

dL

X

Do

- D

j

14

(A-9)

This expression for average wall shear stress can now be used to

define the Fanning friction factor for annular

flow.

F

c

=

Tw,avg

I

(1/2

Py =

dP

I

dL

x Dhy I

(2

PY

(A-IO)

Next, we require this be the same function

of

Reynolds number as

for laminar pipe flow, Eq. A-7.

This

leads to the following equation

for Reynolds number.

N

Re

=

P vD

L

2

1

p.

Dhy)

=

P v

Deq

I p.

A-U)

The various diameters are defined as:

Lamb's diameter:

D

L

=

[0 0

2

+ Dl- (D

0

2

-

Dl Iln D

o

I D

j

]112

seeEqs.A- I and

A-S .....

(A-12)

Hydraulic diameter:

Dhy =Do - D

j

=Outer diameter-Inner diameter A-l3

Equivalent diameter:

Deq

=D

L

2

1Dhy' Concentric

Annulus (A-14)

An average wall shear rate can be defined that

is

consistent with

Eq.

A-9 fOr the averagewall shear stress. Fo r isothermalNewtonian flow,

when the flow is laminar and the fluid has time-independent proper

ties.

Yw

=

[(3 N + I)

14

N] x S v

I

D)

(A-16)

Where

N

is defined

as:

N

= d[ln

T

w

] I d[ln(Sv I D)]

= d[ln(dP I dLxD

14 ]

Id[ln

Sv

I

D

(A-17)

nNn is the slope of a log-log plot of the two variables dP/dL D/4)

and BvID).

This

slope varies with flow rate and rheological

properties.

Fo r

a general time-independent non-Newtonian fluid,

Metzner

Reed

used N to express wall shear stress as a function

of

the Newtonian shear rate at the

wall of

a pipe, Eq. A-4.

T

w

= d P / d L x D / 4 = K S v / D ) N

(A-IS)

In general, both K' and N are functions

of

Two In cases where the

slope

of

the log-log plot is not constant, N is the slope

of

a line that

is tangent at a particular point on the curve for In(dP/dL x D/4)

versus In BvID). When N is a constant and equal to 1, K' reduces

to the Newtonian coefficientof viscosity, and Eq. A-18 reduces to the

form of Eq. A-2 When N is greater than 1, the fluid is a dilatant,

and when N is less

than

1,

the fluid is a pseudoplastic. Hence, as

noted by Metzner

Reed, N is a measure of a fluid's non-Newtoni

an behavior. K' is a measure of the fluid's consistency and increases

as the fluid becomes more viscous.

In

the case

of

a Power-Law fluid, Eq. A-18 becomes:

T

w

= dP I dL x D 14 = Sv I = K ( YW)B (A-19)

Fo r this type

of fluid, N

=

n

=

constant, and Metzner

Reed's

generalized consistency index becomes:

K'

=

K [(3n

+

I) 14n]B

A-20)

Here

the AK without a superscript is simply called the consistency

index.

Note that Eqs. A-19 and A-20 are consistent with Eq. A16

for shear rate at the pipe

wall.

The expression for the generaIized


8/14

8

A NEW MODEL FOR LAMINAR, TRANSmONAL, AND

TURBULENT

FLOW

OF

DRILLING

MUDS

SPE25456

NRe,PL =P

D

V I . w

(3n + 1)

14n] A-24

Now, this equat ion for Reynolds number

can be

simplified even

further and put into the cIassical form for Newtonian flow

by

defining

an Effective pipe diameter which includes the remaining effects of

a Power-Law fluid. Hence, the obvious definition

of

an Effective

diameter for

flow

of a Power-Law fluid through a circular pipe with

diameter D is:

0eff =4n D

I

(3n

+

I), Power-Law pipe flow..

A-25

With this definition of the Effective diameter, the GRE for

flow

of

a Power-Law fluid through a pipe

becomes:

NRe,PL

= P

V D

eff

IIJ.w,app A-26

By comparing Eqs. A-16 and A-25, itmay be seen that the definition

for an Effective pipe diameter

can

be generalized to tbe case of any

time-independent fluid by simply replacing

On

with N in

Eq.

A-25.

Hence, the generalized Effective pipe diameter is defined

as follows.

0eff

= 4N D I (3N + I , Generalized Effective

pipediameter . A-27

Neither Metzner, his students, nor subsequent workers foresaw

any

value in introducing the Effective diameter that is defined here.

As

we shall see, it

is

indeed very helpful in extending their work on non

Newtonian pipe flow to non-Newtonian annular

flow.

Before continuing on to annular flow, an expression is needed for N

that

is

applicable to Herschel-BuIkIey (HB) fluids. SinceN is defined

by Eq.

A-17, a relation between

wall shear

stress and flow rate for

an

HB fluid

is

necessary in order to evaluate N. The required relation

Ship can be found in the book by Govier and Aziz.:IlI

The

resulting

equation for N

is

given below.

lIN = -3 + 'T

w

{ I + m /[m ('T

w

- YS)] + 2('T

w

- YS)

[A(3m +

1)]

+

2 YS/[A 2m

+

I)]} A-28

Z

=

[1-

(Oil D

o

Y]l/

Y

A-33

G = 1

+ Z/2

[(3 -

Z n +

1]

1[ 4 - Z n]

..

A-34

dP I dL = (4 KI DIIy) [8 v G I DIIy]Il A-3S

The authors state that

this

correlation

is

based

on the analytical

solution originally obtained by Fredrickson

and

Bird.t

This

equation

can be

related to Power-Law pipe flow by rearranging

Eq.

A-35 into

the form of Eq. A-19, viz.:

'T

w

=dP

I

dL x Oily

14

=K (8

V

G

I OIly)1l

=

K (8

v

I

Oeff)1l

A-36

Hence, the Effective diameter for laminar Power-Law flow through

a concentric annulus

is

more complicated than simply insertingDeq

into

Eq.

A-25.

It

is defined by

Eq.

A-36 to

be

equal to:

=Oily

I

G =Effective Diameter for non-

Newtoman flow through Concentric

Annuli ... A-37

We

have

designated Eq. A-37

as

being valid for a general non

Newtonian fluid because G

can be

generalized by simple replacing

On

with

nNw

in

Eq.

A-34.

This

Effective

Dianleter

is

a function

of

both the annular geometry and the rheology of the fluid. and it

provides the link between non-Newtonian annular

flow

and Newtoni

an pipe flow.

The

ExIog correlation

can

also

be

applied to flow of an

HB

fluid

by

noting:

T.

- YS

=

[dP/dL]HB

X

DIIy 14 -

YS

== K (8 v/Deff)8l A-38

or

[dP

I

dL]HB x

DIIy

14 =YS +

K (8

v

I

D

eff

Il ..... A-39

Hence, the utility of the ExIog correlation can

be

extended by simply

adding the Yield Stress to the average wall shear stress created by

movement

of

the fluid

and

(1) replacing

On

with

N

in the definition


9/14

SPE 25456

T. REED AND A Pll..EHVARI 9

difference solution

by

Haciialamoglu and Langlinais.' They

baYe

computed the

frictional

pressuregradientgenerated

by

flow

ofan

lI

fluid

through a concentric annulus

for

the following conditions: Do

=

10

in. [25.4 em], D

i

= 7 in. [17.8 em], K = 250 eq. cP,

m

=

0.70, YS

=

51bf/100 fi2 and a pump rate of

200

gpm

[12.6

IJs].

For these conditions, the authors report a calculated pressure

gradient of

0.00870 psiIft.

Using the method outlined herein, a

prcssure gradicnt of

0.0086 psiIft

is obtained. This conclusively

disproves the claim by Hanks that thc Generalized Reynolds

Number cannot be applied to annular

flows.

TURBULENT NEWTONIAN FLOW

Jones and

Leuni

proved that thc equivalent

diameter,

as defined by

Eq. A-14, could also be used in turbulent flow to relate concentric

annuli to circular pipes. They assembled a range of test data and

showed frictional

pressure gradients

in

a smooth annulus could be

correlated by

using

thc Equivalent diameter in thc classical

Colcbrook

e q u a t i ~

for smooth

pipes.

This leads to the

next

question: Can this be extended to non-Newtonian flows?

In

order

to answer this question, we next tum our attention to the work by

Dodge and Metzner on nOD-Newtonian pipe flows.

TURBULENT NON-NEWTONIAN FLOW

Pipe

Flow. Dodgc and Metznefl were able to correlate turb\llent

pipe-flow data for a varicty

of nOD-Newtonian

fluids.

accomplished by using the GRE ofMetzner and Reed and modiJYing

the Colebrook Equation.

They

proposed the following

form

of

Colebrook'sEquation for

all

time-independent fluids flowing through

smooth circular pipes.

1 / F

r

1l2

= (4 /

Nl7S)

log [ N ~ - Nfl)] - 0.40 / N1.2(A-4S)

In

order to apply this equation correctly, the apparent viscosity must

be computed in a particularway. Dodge Metzn,:r emphasized b&t

the shear rate in Eq. A-23 must be based on an unagmary laminar

flow

velocity

that will generate the same

wall

shear stress as the

turbulent flow. Hence, an iteration is required

in

order to obtain the

The relative wall roughness is specified

by

the ratio E

D .

The corresponding equation ofDodge Metzner for turbulent nOD

Newtonian flow, Eq.

A-45, can

be rearranged into

this

same form.

This results in a ncw equation for turbulent friction factors which

combines

nOD-Newtonian

and wall-roughness effects.

l.2

F

r

1l2 =- 4 log [(0.27 E / D

eff

+

1.26

r

/

~

N ~ - Nf2) ] .. . (A-47)

This extended form of the Colebrook equation can be applied to

non-Newtonian

flow

through pipes and concentric annuli by simply

using the

correct

forms

for

the

frictiOD

factor,

the EffectiveDiame

ter, and the Generalized Reynolds number.

There

is

a legitimate question about whether the coefficient for thc

roughness term (0.27) should be a function of N or not. Note that

the Effective Diameter is a function of

N,

and in the case of

pseudoplastics (N < I), a given

wall

roughness, will causea greater

percent increase in the frictional pressure gradient than occurs in

Newtonian flow.

I t

is uncertainwhether this is sufficient to properly

account for the influenceof non-Newtonian fluids onwall roughness

cffects. Additional tests with bentonitc muds

in

rough pipes (E

=

0.0047

in.

[0.119

mm] from water tests) show good agreement

with

Eq.

A-47

up toReynolds numbers of

270,000.

However, additional

tests

with

systematic variations ofmud properties and relative rough

ness are needed beforeEq. A-47 can be verified conclusively. In the

meantime, this equation is proposed for calculation of

frictional

pressure gradients in turbulent non-Newtonian flow through pipes

and concentric annuli with either smooth or rough

walls.

TRANSmONAL FLOW

Geometric

Effect&. Hanks

1

andHanks andPctcrson

lS

havc rcported

experimental measurements of transitional

flow

through concentric

annuli for Newtonian

fluids.

Reference

15

presents experimental

data for concentric annuliwith four different ratios of iDDer-to-outer

diameter (aspect ratio). The reportedvalues ofReynolds numberat

transition are plotted in Fig. 1

Hanks also

included the results from

a transition theory that he first began developing back in the

early


10/14

10

A NEWMODEL FOR LAMINAR, TRANSmONAL,

AND

TIJRBULENT FLOW

OF DRllJ .JNG

MUDS

SPE2S456

similarity analysis properly accounts for geometric effects. As before,

this

idea

can

be extended

to

include non-Newtonian effects

by

using

the Effective diameter, Eq. A-37.

Non-Newtonian

Etl'eet&. In

Hanks 1963 -paper, he used the same

theory as he used for Newtonian

flow

through concentric annuli to

predict transition Reynolds numbers for Bingham Plastics in pipes.

He plotted his results in terms of a critical Bingham Plastic Reynolds

number. TheBinghamPlastic Reynolds number for a pipe is defined

as:

NRe,BP

= D

v /

(Plastic Viscosity) (A-50)

Note that this

is

not a correct measure of Reynolds number effects

because the Plastic VISCOSity PV is coiJstant for a given fluid, and,

in reality, the apparent viscosity will

vary

with she r rate, Eq.

A-43.

Hanks critical values for the Bingham Plastic Reynolds number

are

shown

in

Fig. 3 as a function

of

Hedstrom number.

The critical

Reynolds number is a measure of the flow rate atwhich laminar

flow

ends and transition begins. The Hedstrom number is a measure of

the influence of a fluid s yield stress on the

flow.

For the

c se

of a

Bingham Plastic

flowing

through a pipe, it

is

defined

as:

-He = Hedstrom No. = p

yp l 2

/ (pV)2 ...... A-51

Hanks theoreticalvaluesofcriticalBinghamPlasticReynoldsnumber

appear to agree with measured data up to a Hedstrom number of

about 5,000 as plotted in his original 1963 paper.

z

At higher values,

his

theory diverges from the data,

see

Fig.

3. In

contrast, the solid

curve in this figure

passes

through the experimental data. This curve

was

generated by utilizing the Generalized Effective diameter for a

pipe, Eq. A-27, and the correct apparent viscosity to define a

Generalized Reynolds number,

Eq. A-26.

Next, the new transition criterion requires the critical condition for

any

fluid to occurwhen the product

of

the Fanning friction factor and

the GRE equals 16.1. This generalized transition criterion is designed

to reduce to a critical Reynolds number

of 2100

for Newtonian pipe

flow.

The GRE for pipe

flow is

related to the BP Reynolds number

This agrees with the results ofMetzner for BP fluids and, again,

demonstrates the consistency

of

the

similarity

analysis.

Friction

Factors through the

Transition Regime.

In

a

1977

paper,

Churchill

30

developed a simplemethod for calculating friction factors

through the transition zone

in

Newtonian pipe

flow.

Churchill

devised the following procedure for combining laminar and turbulent

friction factors

in

order to calculate friction factors through the

transition zone.

He

first defined an intermediate term based on the

transitionaland fully-turbulent friction factors; this term

is

designated

F

1

and

is

defined by:

F

1

)-8 =(F1r)-8

+

F

tudl

)-8 (A-55)

F

1

is

then used in a

similar

equation

involving

the laminar friction

factor to compute friction factor through the transition zone.

Ff)12

= F

1

)12 + F

Iam

)12

A-56

This equation

can

then be solved for friction factor at

Reynolds

number or

flow

regime.

Churchill selected the

following

functional form for friction factor

in

the transition zone.

Fir

=

Const. x N

Re

2

A-57

sed on the numerical computations of Wilson and Azad

31

Churchill

chose

a value of 1.42x for the constantwhen the

fluid

is Newtonian. We have extended Churchill s analysis to non

Newtonian fluids

by

replacing the constant coefficientwith a function

of N . An analysis of the friction factor curves of Dodge and

MetzneeZ1

for n

=

I, 0.8, 0.6 and 0.4, Fig. 2, and using the generally

accepted criterion of 2100 for transition in Newtonian pipe flows led

to the

following

equation for this coefficient.

C(N)

=

9.4

x

10-

9 /

[4.767 - 2.167

N]

2

(A-58)

This equation should be used

in

place

of

the constant coefficient

in

Eq. A-57.

In

addition, the Reynolds number

can

be replaced

with

the

GRE.

This leads to the following equation for transitional


11/14

I

Fig. 3 Transition in Bingham Plastic

plpeflows.

I

Thousands

~

I

l

i>

.....

10

8

1.0

0.8

0.8

0.4

. . . . . . .:

. . . .

: ..

;

..:..:.:.:.:.

.. . , . . .

.

.

.

. . . . . .

;

: :..:. .:.:.

. . .

..:..1..

10

4

10

6

HEDSTROM NUMBER

..1.

.......

; ...

1:21

...

-\.

I ~ < l

Fig. 4 - Critical Generalized Reynolds Number

fo r

Btngham Plastic

pipeflows

l-

f

..--

I - -

I

1,.-1-'

I---

FFORE ..

18.1

l

I

~

i-o f:

----

~

I

..

..

I

..

..

H A N K S

1 8 8 3 ) I

..

..

..

I

1800

1500

1200 3

10

2400

2100

2700

3000

I Fig. 2 - Fiction factor. for Power-Law plpeflows. I

0.1

:.. ,.; :..:.: .. 1 :. L:

:.1

:

J

a:

t

if

z

o

~ l

l ~ r h i

...

z

z

if

_ _

. . . .

0.001

I I I I 1 I

II I

I I I I I I II I I I I

I

II

10

2

10

3

10

4

10

GENERALIZED REYNOLDS NUMBER

.......:

. . . . ;

...

;

..:

..

:.:.:.:.

o

z

>=

W

II:

o

W

N

::::i

t::Dt;;

~

. . . . -

.

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

12-1/4 x

5 Annulus

Example Mud Properties

300

< Pumprate

~ 1 0 0

ti

tb 601- . .

,

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

. . .

121 4 x 5 Annulus


.....: : ; : : ; : :

: .

1

0.2

0.8

-0 .6

0.4

Fig. 9 - EFFECTS'OF

SHEAR

I tATE

EXPONENT

ON

EFFECTIVE

DIAMETER

OF

ANNULUS

i>

d

....

21

...

19

6 23 8 Annulus

MMH Mud System

, to

7 9

11

13

16

17

PUMPRATE 10 0 IGPM

5

0

..................... , , , .

. . . . . .

......;. .. -; :-

.:.

~ : : : ; ..

-:

:-

.:.

~ : : ; : :-

.. ..

;..; -:.; .: :.

0

, I I I , , , I ,

3

F i g - E FF EC TS O F PUMPRATE

O N

EFFECT IVE

V IS C O S IT V IN

ANNU LU S

0.1\

: : :

: S;:: ::

i: :; :; :;: i

:;:;:;:

:::::; ::

:;:

:

:;:

:;: >:;;;

:::::

:;:

:

:;: :;:; :;:

:::;

c:J

z

i

z

a:

e

IL

Z

o

ti 0.01

iii

Yo

1

.; :- .;. : :.: ;. :-

.. -:..

: -:. : .: :.

12-1/4 x 6 Annulus


t

. . . . . ; ~ ~ .:. .: : : ;. ,:::: ::: : J ~ ~ ~ . O T _ H J ' : -- : '

0.11

.?S.;.,

'..' l

:

:

:: :

:

. : ::

:

::

:::

:::::::: :::::: ::

:::

: ::: ::: :::: :::::::

: :::

:::::

:

:

::::

: ::

0 , I

J

I I I I I

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

N GENERALIZED SHEAR RATE EXPONENT

a:

z

0

ti 0.01

iii

Yo

c:J

z

i

z

it

I I I

5

0.001 2

3

10

4

10

10 10

I I , 6

0.001 ' 2

3

104 10

10 10



FIG - FR ICT ION FACTORS FOR ANNUL I

W ITH SMOOTH AND ROUGH WA LL S

FIG. 2 - COMPARISON O F

MODEL

WITH TEST

DATA FO R MMH IN A ROUGH ANNULUS

.


14/14

~

T LE

P R METERS

FOR TR NSITION ZONE IN ROUGH 2 Yl NNULUS

- - - - ----

MUD FANNING

M U D F L O ~ E

VELOCITY FRICTION GENERAL.

S H R ~

RELATIVE

FPGRAD

(GPM

FfIMIN)-

N FACI OR REY. NO.

W GRE FLOWTYPE

l/SEe

ROUGHNESS (In.H,llIFt)

1860. 364.5 .3609 .6151E-02

26042E+03

16.02

LAMINAR

186.2

.015627

.4034

1870.

366.5

.3617 .6098E-02 26284E+03

16.03

LAMINAR 187.1

.015607

.4043

1880. 368.4 .3625 .6046E-02 26S29E+03

16.04

LAMINAR 188.0

.015588

.4051

1890.

370.4

.3634 .5996E-02 26781E+03 16.06

LAMINAR

188.9 .015567 .4060

1900.

3724

.3643

.5947E-02

27042E+03

16.08

LAMINAR 189.9 .015545

.4070

1910.

374.3

.3653 .5901E-02 27314E+03

16.12

lRANSIT

191.1 .015522 .4081

1920. 376.3

.3664

.5857E-02 27603E+03 16.17 lRANSIT 1924 .015497 .4093

1930.

378.2 .3679 .5819E-02 27947E+03

16.26 lRANSIT 194.1 .015462

.4110

1940.

380.2 .3699 .5790E-02 28359E+03 16.42 lRANSIT

196.4

.015417 .4131

195 3822

.3741

.5797E-02

29083E+03 16.86 lRANSIT

201.4 .015321

.4179

1960. 384.1

.5574 .1061E-Ol

6.0752E+03

64.46 lRANSIT 634.7 .012561 .7728

1970. 386.1

.5595 .1060E-Ol

6.1540E+03 65.25

TIJRBULENT

644.6

.012539 .7802

1980. 388.0

.5617 .1060E-Ol

6.2329E+03 66.05 lURBULENT 654.7 .012517 .7876

199

390.0

.5637

.1059E-Ol

6.3121E+03 66.84 lURBULENT 664.8 .012496 .7950

2000. 3920

.5658 .1058E-Ol

6.3915E+03

67.63

lURBULENT 674.9 .012476

.8024

2010. 393.9

.5678 .1057E-Ol

6.4711E+03 68.43 lURBULENT 685.2 .012456 .8099

2020.

395.9

.5699

.1057E-Ol

6.5509E+03

69.23 lURBULENT

695.5

.012436 .8175

2030.

397.8

.5718 .1056E-Ol

6.6310E+03

70.03

lURBULENT

706.0

.012417 .8251

2040. 399.8

.5738

.1055E-Ol

6.7112E+03 70.83

1URBULENT

716.5 .012398

.8327

2050. 401.8

.5757 .1055E-Ol

6.7917E+03

71.63

TIJRBULENT

727.1 .012379

.8403

2060.

403.7

.5776

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a new model for laminar, transitional, and turbulent flow of drilling muds

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