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Modelo de flujo de fluidos

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Society
of
PetroleumEngineers
SPE25456ANewModelforLaminar Transitional andTurbulentFlowofDrillingMuds
T.O.
Reed,
ConocoInc.
and
Pilehvari
U.
ofTulsa
SPE
Members
Copyright
1993,
Societyof
Pf1.\roleum
Engineers
Inc.
Thispeper
was
preparedforpresentation
at
theProductionOperationsSymposiumheld
In
OklahomaCity
OK,
U.S.A. March21 23
1993.
Thispaper
was
selectedforpresentation
by
an
SPEProgram
Commilleefollowingreviewofinformationcontained
In
an
abstractsubmitted
by
the
author s .
Contents
of
thepaper
as
presented
have
not
beenreviewed
bytheSocietyofPetroleumEngineers
and
are
SUbject
tocorrection
by
the
author s .The
material
as
presented doesnotnecessarilyreflectany
pos~lon
oftheSocietyofPetroleumEngineers
Its
officers ormembers.
Papers
presented
atSPE
meetings
are
subject
to
publicationreviewbyEditorialCommitteesoftheSocietyof
PetroleumEngineers.Permission
to
copy
Is
restricted
to
an
abstractofnot
morethan300words.
illustrations
may
not
be
copied.
The
ebstrect
shouldcontainconspicuousacknowledgment
of
w r~
andbywhom
thepaper
is
presented.WriteLibrarian
SPE,
P.O.
Box833836,
Richardson
TX75083-3838,
U.S.A.Telex
163245SPEUT.
STR CT
Theconcept
of
an
Effective
diameter
is
introducedforthe
flow
of
drillingmudsthroughannuli.
This
new
diameter
accountsforbothannulargeometryandtheeffects
of
anon-Newtonianfluid.
It
providesthelinkbetweenNewtonianpipe
flow
andnon-Newtonian
flow
throughconcentricannuli.Themethod
is
validin
any
flow
regimeandcan
be
usedtodeterminewhethera
DOn-Newtonianflow
is
laminar,transitional,
or
turbulent.
n
analyticalprocedure
is
developedforcomputingfrictionalpressuregradientsin
all
three
flow
regimes.
The
analysis
also
quantifiesbow
flow
transition
is
delayed
by
increasingtheyieldstress
of
afluid.
In
addition,
it
is
shownthattransitioninanannulus
is
delayed
to
higherpumpratesastheratio
of
innertoouterdiameterincreases.Furthermore,themethodaccountsfor
wall
roughnessanditsaffectsontransition81
and
turbulentpressuregradientsfornon-Newtonian
flow
throughpipesandconcentricannuli.
Finally
themethodrunsona386
P
inonlya
few
seconds.
INTRODUCTION
ThestandardAPImethodsfordrillinghydraulicsassumeeitheraPowerLaw
or
aBinghamPlasticrbeologymodeL
In
reality,mostdrillingmudscorrespondmucbmoreclosely
to
theModifiedPowerLaw
or
Herschel-Bulkleyrbeologicalmodel.
This
distinction
is
particularlyimportantfortheannulargeometriestypical
of
normaldrillingconditionswhereshearratesareusually
low
andthePowerLawunderestimatesandBinghamPlasticmodeloverestimatesfrictionalpressuredrops.
This
paper
also
showsthatthese
two
classicalmodels,respectively,underestimateandoverestimatepumpratesrequiredfortransitionfromlaminat
to
turbulent
flow.
Althoughthere
have
beenanumber
of
paperspUblishedonthelaminar
flow
of
yield-pseudoplasticsthroughannuli/-Jnoneclaim
to
beuniformlyvalidforlaminar,transitional,andturbulent
flow.
Thiscanbeaccomplished
by
introducingtbeconcept
of
an Effective diameterwhichaccounts.forbothannulargeometryandtheeffects
of
anon-Newtonianfluid.
This
diameter
also
enablestheinclusion
of
theeffects
of
wall
roughnessonfrictionalpressuregradientsandtheprocess
of
flow
transition.Furthermore,theresultingmodelcanbeusedtodeterminewhether
flow
of
anon-NewtonianfluidthroughReferencesandillustrationsatend
of
paper.apipe
or
concentricannulus
is
laminar,transitional,
or
fully
turbulent.Aderivation
of
themodel
is
presentedintheAppendix.Someadditionalbackgroundandresultsfromthemodelaregiveninthefonowingdiscussions.
NEWTONIAN
FLOW
RelatioDShipBetween
Pipe
and
nnular
Flows.
A
considerablenumber
of
equivalent diameters
have
beenproposedoverthe
years
for
flow
throughconduitsother
than
circular
pipes.1O-1z
Thepurpose
of
definingsuchadiameter
is
to
introducedefinitions
of
friction
factor
andReynoldsnumberthat
will
enableapplication
of
thewenknownrelations
for
pipe
flow
toothergeometries.
In
particular,theobjectiveisto
be
able
to
calculatewhatthefrictionalpressuredrop
will
beforagivenfluidataparticular
flow
rate.
h
shownintheAppendix,thefrictionfactorforaconcentricannulusshouldbe
based
ontheHydraulicDiameter,
Dhy>
which
is
simply
the
differencebetweentheinnerandouterdiameters.TheReynoldsnumbershouldbe
based
onan
~iv l nt
Diameterequal
to
thesquare
of
Lamb sDiameter,D
L
divided
by
Dhyt
seeAppendixEqs.
A.12,
13
and
14.
Whenthefrictionfactorand.Reynoldsnumberaredefinedin
this
manner,theclassicalrelationsforNewtonianpipe
flow
e.g.,theMoodyDiagram,canbeapplied
directly
to
concentricannuli.
In
fact,JonesandLeung
13
haveproventhat
these
definitions
also
apply
to
fully-turbulent
flow
throughanannulus.
This
was
demonstrated
by
showingthatavariety
of
experimentaldataforconcentricannuliagreewiththeColebrookEquationforturbulentpipe
flow
which
is
equivalenttothe
fully
turbulentand
hydraulica1ly
roughportions
of
theMoodyDiagram.FlowTransition
in
Annuli.
In
additiontobeingapplicableinthelaminarandturbulentregimes,thesesamedefinitionscanalso
be
usedtopredictthe critical Reynoldsnumberatwhichlaminar
flow
endsandtransitiontoturbulencebegins.Prof.Hanksand
his
students
have
publishedexperimentaldata
on
flow
transitioninconcentricannuli.
I
4-15
Some
of
theirdata
is
shownin
Fi
1.
Thedashedcurve
is
theresults
of
predictionsfromatransitiontheory
469
2
A
NEW
MODEL
FOR
LAMINAR,
TRANSmONAL,
AND
ruRBULENT
FLOW
OF
DRILLING
MUDS
SPE2S456
developed
by
Hanks
in
the
early1960s.
7,1 17
The
solidcurvecomesfromsimplysetting
the
Reynoldsnumber,
based
on
the
EquivalentDiameter
of
Jones
and
Leung,equal
to
2100.AReynolds
number
based
only
on
Lamb's
Diameter
is
used
for
the
ordinatein
order
to
displayhow
the
criticalvalues
vary
withdiameterratio.
It
is
readily
seen
thatthemuch
simplertransitioncriterionprovides
better
agreementwith
Hanks
experimentaldata.
The
newcriterionshowsthattransitionfromlaminarflowisdelayed
to
progressivelyhighervelocities
or
pump
rates
as
diameterratioincreases.
NON-NEWTONIAN
FLOWPipe
Flow.
Prof.Metzner
and
hisstudentsreported
some
~ion rin
workin
the
19508
on
non-Newtonianflowthrough
pipes. l-Z1
TheirexperimentswithPower-Lawfluidsinpipesprovidedthefirstcleardefinitions
of
how
the
frictionfactorvarieswithReynolds
number
inthelaminar,transitional,
and
turbulentregimes.Theyfoundthatadecreasingpower-lawexponentdelaystransition
to
higherReynoldsnumbersandshifts
the
turbulentfrictionfactorsdownward.
This
isadirectresult
of
increasingdegrees
of
the
phenomenoncalled
wshear
thinning.
In
addition
to
the
testdata,
Metznerand
hisstudents
also
developedanovelanalysis
that
provideda
way
to
generalizetheirresultsforPower-Lawfluids
to
all
time-independent,non-Newtonianfluids.Theysimplydefined:
Wall
Shear
Stress
=
K
x(Newtonian
Shear
Rate)N(1a)
or
usingstandard
symbols
,N
T
w
=
K
x
8
v/
D
ã
1b)
From
thisequation,itfollows
thatthe
definition
of
the
exponent
WNW
is:
N
=
d(ln
Twl
/d(ln(8
v/
D)](1e)
Theyshowedhow
the
twoparameters
K'w
and
WNW
can
be
applied
to
Power-Law
PL)
fluids
and
BinghamPlastics(BP).
The
Appendixprovides
the
details
on
how
this
approach
can
also
be
applied
to
aHerschel-Bulkley
(BB)
fluid,which
is
essentiallyacombination
of
the
PL
and
BP
rheologicalmodels.
The
Metzner
group
alsodeveloped
the
followingrelationforwallshearrateinapipeforageneralfluidwithtime-independentproperties.
Generalized
PipeShearRate:
Yw
=
(3N
+
1)8v/
4N
D)
ãã
(2)
This
expressionledusto
the
followingconclusion.
n
Effectivewdiametercan
be
defined
so
that
the
generalizedshear
rate
can
beput
into
thesame
form
as
theshearrate
forNewtonianpipe
flow,
viz.,
(8v/D).Using
this
logic,
the
Effective
Diameter
forgeneralizednonNewtonianpipeflow
is:
WElTective
w
PipeDiameter
=
D
eff
=
4N
D/ 3N
+
1)....(3)
The
wEffective
w
diameter
for
non Newtonian
pipe
flow
is
the
diameter
a
circular
pipethat
would
havetheidentical
pres-
sure
drop
for
flow
a
Newtonianfluid
with
viscosityequalto
the apparent
viscosity
and
the
same
average
velocityas
thenon Newtonian
flow.
Note
that
this
diameterisless
than
the
physicaldiameterforpseudoplastics
N
<
1)
and
isgreater
thanthe
physicaldiameterfordi1atantfluids
N
>
1).
As
will
be
shown,theEffective
Diameter
makes
iteasier
to
relatenon-NewtonianandNewtonianflows.
The
advantage
of
suchaconnectionisthatthewell-establishedfrictionfactorrelationsforNewtonianflowscanthen
be
applied
to
non-Newtonianflows.
This
greatlysimplifies
the
task
of
computingfrictionalpressure
drops
forsuchflows.
The
remainingvariablethatisneeded
to
define
the
GeneralizeReynolds
Number
is
the
apparent Newtonianviscosity.
This
parameterisdefined
by:
App.Viscosity
=
ShearStress/Shear
Rate
@
Wall(4a)
1J.w,app
=
K
(8
v/
D)N/
Yw
.......................ã...........
u
(4b)
Withthesedefinitions,
the
GeneralizedREynoldsnumber
GRE)
isdefinedas:
NRe,O
=
vDeffl
1J.w,app
ããããããããããããããããããããããããããããããããããããããããããã
ãã
(5)
This
expresses
the
Reynolds
number
fornon-Newtonianpipeflowin
thesame
algebraicform
as
foraNewtonian
fluid
Metzner
&
Reed gshowed
thatthe
GRE
isrelated
to
laminarfrictionfactor
by
thesame
classicalequationforNewtonianflow,
viz,
c
=
16/
NRe,o
,LaminarNon-Newtonian
Flow......
(6)
These
authorsshowedthatexperimentalpipeflow
data
foravariety
of
non-Newtonianfluidsfollowedthisrelationship.
Later
experi.mentswithPower-LawfluidsinturbulentpipeflowledDodge
Metzner3l
to
modify
the
classicalColebrookequationforturbulentfrictionfactorsinNewtonianpipeflow,
see
Eq.
A 45
This
was
necessaryin
order
to
correlate
thedata
because
the
shear-rate
exp0-
nent
was
found
to
be
asignificantparameter.
In
particular,frictionfactors
decrease
withdecreasingvalues
of
the
Power-Lawexponent.Figure
Z
shows
the
correlations
Dodge
Metznerdevelopedfor
laminar
and
turbulentflow.
The
transitionzoneswere
not
correlated
by
theseauthors,
but
theirexperimental
data
follows
the
indicatedcurvesbetween
theend
of
laminarflowand
the
beginning
of
fully-
turbulentlow.
As
describedin
the
Appendix,a
new
correlationfor
the
transitionzones
bas
been
developed
based
onthe
experimental
data
of
Dodge
&
Metzner.
The
curvesin
the
transitionzones
are
from
this
newcorrelation.
The
generalizedshear-rateexponent
Nil
can
be
used
to
extendthe
data
of
Dodge
MetznerforPower-Lawfluids
toothernon·
Newtonianfluids.
In
partiCUlar,
Dodge
Metznerderivedtheappropriateequationsfor
the
GRE,
shearrate,andapparentviscosityforflow
of
a
BP
throughpipes.TheirworkisextendedtoHerschel-Bulkleyfluidsandconcentricannuli.in
the
Appendix.However,beforediscussingannularflow,itisappropriate
to
firstdiscuss
some
resultsfrom
the
newtransitioncriterionforfluidswithayieldstress.
Hanks
and
Pratt
J
proposedamodelforflow
of
BinghamPlasticsthroughpipes.Their
paper
also
includesatransitioncriterion.Theychose
to
base
theircriterion
on
the
so-calledBinghamPlasticReynolds
numberandtheHedstrom
number.Thesetwodimensionlessparameters
are
definedin
the
Appendix,
see
Eqs.A-50and
A 51
Unfortunately,
the
BP
Reynolds
number
is
not
alegitimatemeasure
of
Reynolds
number
effectssimplybecausethePlastic
Vis·
cosity
(PV)
isaconstant
and
does
not
vary
withshearrateastheapparentviscosity
does.
Furthermore,
theHanks
and
Pratt
criterion
doesnot
agreewithexperimental
data
at
highervalues
of
theHedstromnumber.
J
This
disagreementisshowninFig.3.Otherinvestigatorshave
also
notedthisdisagreement.
Z1 22
In
contrast,thesolidcurvein
Fig.
3
is
based
on
thenewtransitioncriterionwhichrequires
the
criticalconditionforanyfluid
to
occurwhen
the
product
470
SPE
25456
T.REEDAND
PILEHVARI
3
of
theFanningfrictionfactor
andthe
GRE
equals16.1.
(As
may
be
seenfrom
Eq.
6,
this
product
is
16
in
Jaminar
flow,
and
it
exceeds
this
valuewbentransiqonbegins.)
This
generalizedtransitioncriterionisdesigned
to
reduce
to
acriticalReynolds
number
of
2100forNewtonianpipe
flow,
see
lastsection
of
Appendix.
The
underprediction
of
thedata
by
the
Hanks
and
Pratt
criterion
is
even
more
apparentwhencriticalvalues
of
the
GRE
are
plottedasafunction.
of
Hedstrom
number,
Fig.
4.
Here
itbecomesclearthattbe
HanksandPratt
criterionpredictsdecreasing
flow
rates
for
transitionas
the
YieldPoint
(YP)
increases.
This
certainlyviolatestbeexpectedtrend.
The
newgeneralizedcriterionshow
the
expectedtrend,i.e.,higher
pump
rates,
are
required
to
achievetransition
as
tbe
yP
of
a
BP
increases.
It
is
instructive
to
plot
the
frictionfactorsfor
BP
pipeflowthroughall
three
flowregimeswith
Hedstromnumberas
aparameter.This
is
sbownin
Fig.
5.
Thisfigureshowsbow
an
increasing
yP
delaystransition
to
higher
pump
rates
and
also
extends
tbe
transitionzoneoveraprogressivelywiderrange
of
pump
rates.
For
example,wben
tbe
Hedstrom
number
is500,000,
the
flawis
not
fullyturbulentuntilaReynolds
number
of
100,OOO
This
compareswithacommonvalue
of
about3,000forNewtonianpipeflow.
It
is
particularlyimportant
to
observe
that
all
of
the
curvesfor
~ ns t on
eventuallymergeinto
tbe
turbulentcurveforNewtonian
flow.
This
occursbecause
the
yP
becomesprogressivelylesssignificant
as
sbear
rate
increases.Hence,
the
curveshave
the
expectedasymptotes..
There
is
one
other
point
to
noticefromcomparingFigs.5
and
3.
A
BP
bas
a
sbear-rateexponent
of
1;
wbereas,a
PL
fluidgenerallybasanonunityexponent.This
is
the
basicreasonwby
the
frictionfactorcurvesfordifferentvalues
of
the
sbear-rateexponent
do
not
merge
and
remaindistinct
for
all
Reynoldsnumbers.
In
tum,
this
impliesthataHerschel.Bulkleyfluid
will
exhibitsomewbatdifferentbehaviorsinceitcombines
tbeBP
andtbe
PL
rbeologicalmodels.
In
particular,
one
caninferthatayieldstress
will
delayandextend
the
transitionzone,
andtbe
transitionalvalues
of
frictionfactor
will
eventuallymergewith
the
fully-turbulentcurvefora
PL
fluidwith
an
exponentequal
to
whateverappearsintbeHerscbel-Bulkleymodel,
see
Eq.
7below.
At
tbispoint,itisappropriate
to
introduce
some
viscometer
data
forarepresentativedrillingmud.Sometypicalrotating-cupdata
are
shownin
Fig.
6.
Dial
readingsfor
six:
rotaryspeeds
are
given.
The
twoparametersin
the
BP
model
YP
and
PV and
the
PL
model
K
and n )
are
determinedaccording
to
standard
API
procedures
by
fittingthese
two
rheologicalmodels
to
the
300
and
600
rpm
data.
In
contrast,
the
threeparameters
of
the
HB
modelallowagoodfit
to
all
six
datapoints.Sinceannularflowshearrates
are
normallybelowthe100
rpm
(170
sec-I)
viscometerdata,
it
is
important
to
haveamodelwhichisvalid
at
low
shear
rates.
On
theother
hand,
the
HB
modelcanalso
be
used
to
estimatefrictionalpressuredrops
and
flowregimesforflowthrough
drill
pipeswhereshearrates
are
high.
The
HB
parametersfor
the
particular
data
of
Fig.
6are:
T
=
YieldStress
K(Shear
Rate -
:::
14
0.281
,y o.m
Ibf/l00
ft2
[0.4788
Pal
(7)
Theseparameters
will
be
usedforillustrationpurposesinFigs.7
tbru
11.
Aweight
of
12
ppg[1,438
kglmJ]
will
also
be
assumedforthisexamplemud.Figure7presentsfrictionfactorsthrough
the
transitionzonefor
the
example
mud
flowingthroughpipes.
The
PL
modelleads
to
transition
at
lower
pump
rates.Initiation
of
transitionisdelayed
to
higherpumprates
based
onthe
HB
model,
but
fully-turbulentflowoccurs
at
about
thesame
GRE
as
for
tbe
PL
case.
TheHB
curveforfullyturbulentflowdoes
not
mergewitb
the
PL
curve
at
higherReynoldsnumbersbecause
the
sbear
rate
exponentfor
the
PL
modelisonly
0566.
The
resultsfor
theBP
modelis
as
expected
i.e.,transitionisdelayed
tbe
most
by
using
this
model.
Furthermore,
the
extent
of.
thetransitionzoneisexpandedsignificantly,
and
the
GRE
is
an
order-of-magnitudelargerbeforefully-turbulentflowisachieved.
In
order
to
relatetheseresults
to
atypicalfieldapplication,considerflowthroughastandardweight5-incb
drill
pipewith
an
ID
of
4.276incbes.Fu1Iy-turbuientflow
is
achieved
at
a
GREof
3800
and
a
pumprate
Of
270
gpm
[17.0
lis]
using
the
PL
model.
The
correspondingresultsfor
theHB
modeloccurs
at
a
GRE
of
4100
and
a
pump
rate
of
310
gpm
[19.6
lis].
In
contrast,
tbe
BP
modelleads
to
aprediction
of
fully-turbulent
flow
at
a
GRE
of
22,300anda
pumprate
of
670
gpm
[42.3
Urn].
(Note:
The
analysisdoesnotaccountfor
the
effects
of
restrictions
at
tooljointswhicbcouldcauseflowtransition
to
begin
at
lower
pump
rates.)
It
is
also
of
interest
to
compare
the
frictionalpressuregradientsforarealistic
pump
rate
of
600
gpm[37.85
lis].
The
resultsfor
the
PI..,
HB,
and
BP
models,respectively,
are
0.085,0.099,
and
0.104psilft[22.62kPa/m].
The
relativeclosevaluesfor
the
HB
andBP
fluids
are
aresult
of
the
HB
beingfullyturbulent
andthe
BP
beingintransition
at
this
pump
rate.
Again
if
tooljointscausedtransition
to
occur
at
lower
pump
rates
andthe
BP
wereinfully-turbulentflow
at
600
gpm
[37.85
lis],
thenthe
differencebetweenpressuredropsfor
the
HB
and
BP
would
be
larger.
Annular
Flow.
The
flowmodelcan
be
extended
to
concentricannuli
by
altering
tbe
definition
of
the
Effective diameter
to
include
the
effects
of
the
differentgeometry.
The
Effective diameterforaconcentricannuli
is
equal
to
DJIy
IG.
The
Effective
diluneterfor
non ewtonianflow
through
a
concentric
annulus
isthe
dia
meter
a
circular
pipethatwouldhavetheidentical
pressuredrop
forflow
a ewtonianfluid
with
a
viscosity
equalto
the
effective viscosity which
is
basedonthe
average
wall
shear
rate
in
t
annulus
and
has
a
velocity
equaltothenon- ewtonianannularflow
velocity.
The
an
function
is
based
on
acorrelation
of
the
analyticalsolution
by
Fredrickson
and
Bird forflow
of
aPower-Lawfluidthroughaconcentricannulus.
This
functionisdependent
on
bothtbe
ratio
of
inner
to
outer
diameters
andthe
shear-rateexponent.
The
function
is
definedin
the
Appendix
by
Eq.
A-34.
This
correlationcan
be
applied
to
other
fluids
by
using
the
generalizedexponent
N ,
Eq.
1c,
inplace
of
the
usual
PL
exponent n.
The
details
are
givenintheAppendix
undertbe
beading Non-NewtonianAnnularFlow.
The
followingdiscussion
will
focus
on
some
resultsforatypicalfieldsizeannulus.
A
12-1/4
x
5 [31.12
emx
12.7
em]
annulus
is
selectedforillustrativepurposes.
The
same
mud
rbeology,
Eq.
7,andaweight
of
12ppg[1,438
kg
m
J]
will
again
be
used
forcomputingfrictionalpressuregradientsthrough
the
annulus.
It
is
pertinent
to
note
that
the
weight
of
the
mud
does
not
affectlaminarfrictionalgradients,
but
weightdoesinfluencetransitional
and
turbulentgradients.Figure8sbowsbow
the
ratio
of
tbe
generalizedexponent
N
over
the
shear-rateexponent
Om
varieswith
shear
rate
for
the
example
HB
fluid.
As
may
be
seenfrom
Eq.
7,
m
bas
the
numericalvalue
of
0.792.Figure8showsbow N approacbes m
as
wall
sbear
rateincreases.Thisis
the
expectedtrendbecause
the
influence
of
theYieldStress(YS)
on
T
w
becomesprogressivelylesswithincreasingsbearrate,compare
Eqs.
1band
7.
In
the
limit
of
infinitesbearrate,
N m
Figure8
also
identifies
tbe
initiation
of
transition
at
1910gpm
[1205
lis],
and
fully-turbUlentflowbegins
at
2010
gpm
[126.8
lis].
This
simplytellsus
thatthe
flowislaminarover
the
range
of
pump
ratesthat
are
normally
used
in
the
field.Unfortunately,
the
~o l
does
not
tellusbow
drill
stringdynamics
may
influenceflow
471
4
ANEWMODELFORLAMINAR,
lRANsmONAL,
AND
TURBULENTFLOWOF
DRTI..LING
MUDS
SPE2S456
transition.
is
reasonableto
expect
the
energy
addedtothe
flowby
theagitation
may
causetransitionat
lower
pumprates.Anotherparameter
in
themodel
is
the
ratio
of
Effective
diameter
over
theEquivalentdiameter.
This
ratio
partia1Iy
removes
geometric
effects
and
highlights
thenon-Newtonian
effects.
ipre
shows
this
ratioapproaches
unity
as
N
goes
toone.
This
is
the
expected
limiting
condition
for
Newtonian
flow.
SinceFig.
8
shows
that
N
increases
with
pumprate,the
Effective
diameter
also
increases
with
pumprate.
Thishas
theneteffectof
reducing
themagnitudeof
the
increase
in
frictional
pressuregradientaspumprate
is
increased,
i.e.,
the
fluid
is
shear
thinning.
This
leads
tothequestion,whathappenstothe
apparent
viscosityas
pumprate
increases?
An Effective
viscosity
is
defined
in
a
man
neranalogousto
pipe
flow.
The
primary
difference
for
anannulus
is:
the
Effective
viscosity
is
defined
by
theaverage
wall
shearstress
acting
ontheinnerandouter
walls
and
divided
by
the
average
wall
shearrate,seeEq.
A-43.This
viscosity
is
presented
in
Fig
10
as
a
function
ofpumprate.
continuouslydecreases
with
pumprateand.
has
anoticeabledropattransition
from
laminar
toturbulent
flow.
This
is
caused
by
the
rapidlyincreasing
shearrate
as
transition
occurs.
Figure
11
presents
friction
factors
as
a
function
o
theGRB
for
the
example
annulusandmudproperties.Forsmooth
walls,
theincrease
in
friction
factorthroughthetransitionzone
is
small.
s
indicated
inFig.8,
thetransitionzone
occurs
betweenpumpratesof
1910
and
2010
gpm
[120.5-126.8
I.Js].
An
example
oftheeffectsofrough
walls
is
also
included
inFig.
11.
t
may
be
noticedthattheinitiationoftransition
occurs
atthesameGRB
This
is
consistent
with
theexperimentaldataofNikuradse
for
Newtonian
pipe
flow,
see
Schlichting.
24
s
indicated
in
the
figure,
transitiontoturbulent
flow
takes
place
overasmallerrangeofpump
rates,
1910
to
1970
gpm
[120.5
to
124.3
I.Js],
andthe
fully
turbulentpressuredropsare
approximately
70
percentlargerthanthecorre
sponding
turbulent
friction
factors
for
smooth
walls.
The
relative
roughness
for
this
illustration
is
approximately0.01.This
is
basedon
dividing
theabsoluteroughness
height
by
the
Equivalent
diameter,Eq.
A-14.
However,
thecorrect
definition
of
relative
roughness
fornon-Newtonian
flowsis
theratioofabsolute
roughnessover
the
Effective
diameter.Hence,
relativeroughness
varieswith
the
fluid
and
the
shearrate.Table1presenta
list
ofsomeoftherelevantparametersthroughthetransitionzone
for
the
rough-wallcase.
t
can
beseenthatthe
relative
roughnessdecreases
as
pumprate
increases.
Theauthors
believe
thesearethe
first
computations
for
theeffectsof
wall
roughness
onnon-Newtonianpressuregradients
in
thetransitional
and
turbulent
regimes.
(Note:pressuregradientsare
given
in
unitsof
inches
ofwaterper
foot;
theseunitsareequalto
0.0361
psiIft
or
0.8164kPa/m.)
Finally,
Fig.12
comparesthe
model
with
testdata
for
a
Mixed
Metal
Hydroxides
MMH
mud
system
flowing
througha
5.023
x
2.375 [12.76
cm
x
6.03
em]
annulus.Thetests
were
conducted
in
Amoco s
Drilling
Hydraulics
Test
Facility
by
personnel
from
thePetroleum
Engineering
Department,Tulsa
University.
Pressuredrops
were
measuredoveralengthof
50
ft[15.24
m].
Pump
ratj:swerevaried
from
50
to
500
gpm
[3.16
to
31.55
I.Js].
Experimentaldata
for
a
high
and
a
low
rheology
MMH
fluid
are
included
inFig.
12
Thecorrespondingrotating-cupviscometer
readings
atthe
six
standard
speeds
are:
327,57.3,61.5,63.0,66.5
and
25,13.5,18.3,220,28.0.
These
values
aretheaverageof
readingsfor
fourdifferent
samples
taken
duringthetimeofa
tesL
s
indicated
inFig.
12,
the
high-rheologyfluid
appearsto
have
been
in
the
laminar-flow
regimefor
theentirerangeofpumprates.The
model
predicts
transitionathigherpumpratesand
friction
factors
that
continuously
decrease
throughthetransition
zone
According
to
the
model,
the
flow
is
not
fully
turbulent
until
aGRBof
18,000
andapumprate
of
77JJ
gpm
[45.42
I.Js].
The
predicted
values
ofturbulent
friction
factorsare
nearly
constant
for
a
wall
roughness
of
0.00018
in
[0.0046
mm].
The
above
valueof
wall
roughness
gave
good
agreement
between
predictedturbulent
friction
factors
andexperimentaldata
for
the
low-
rheologyfluid.
Unfortunately,
water
teststodetermine
wall
roughness
forthe
annuluswerenotperformed.
However,
Fig.
12
shows
encouraging
agreement
betweenpredictionsfor
the
low
rheology
fluid
andthecorrespondingdatathrough
u
three
flow
regimes.
Additional
tests
were
conducted
with
bentonitemuds
flowing
throughrough
pipes.
n
these
cases
wall
roughness
was
determined
via
watertests,
andpredictions
from
the
model,
using
the
experimental
values
of
wall
roughness,
agree
v ry
closely
with
thetestdata
for
u
three
flow
regimes.
This
data
covers
aGRBrangeof
500
to
270,000
and
includes
relative
roughnessvalues
upto
0.002.
Becauseof
spacelimitations,
these
non-Newtonian
pipe-flow
dataarenotpresentedhere.
CONCLUSIONS
1.
A
newanalysis
has
been
developed
for
non-Newtonian
flow
through
pipes
andconcentric
annuli.
Themethod
is
based
oil
relatingnon-Newtonian
flows
to
Newtonian
flows.
The
advan
tage
is
that
well-established
results
or
Newtonian
flows
canbe
applied
to
non-Newtonian
flows.
2The
Effective
diameter
is
a
key
conceptofthe
method.
t
accounts
for
bothgeometricand
non-Newtonian
fluid
effectsonfrictional
pressuregradients
in
pipes
and
annuli.3.
Resultsagree
with
finite-difference
computations
for
laminar
flow
ofa
Herscbel-Bullcley
fluid
through
concentric
annuli.4.
The
analysis
is
valid
for
the
laminar,
transitional,
and
fully
turbulent
flow
regimes.
Themethodincorporatesa
new
transi-
tion
criteriathataccountsfora
delay
in
flow
transition
with
increasing
ratiooftheinner-to-outerdiameters
in
concentric
annuli.
This
agrees
with
experimentaldatareported
in
theliterature.
5.
Model
predictionsof
critical
Reynolds
numbersagree
with
published
data
forBPpipe
flows.
6.
The
Herschel-Bulkley
rheolOgical
model
is
used
which
includes
Newtonian,
Power-Law,
and
BinghamPlastics
asspecial
cases.
This
IDOre
accurate
rheological
model
is
equally
valid
for
predictions
ofpressuregradients
in
drill
pipes
and
the
large
annuli
typical
of
full-scaJe
ri ling~
7.
Whenusing
thesameviscometerdata,results
fromtheanalysesshow
transition
from
laminar
toturbulent
flow
occurs
athigherpumpratesthan
for
a
Power-Law
fluid,
but
signifi
cantlylower
than
thecorresponding
BinghamPlastic
fluid
8.
TheColebrookequationforturbulent
friction
factors
has
been
extended
so
that
itapplies
to
non-Newtonian
flows
through
pipes
and
concentricannuli
with
smoothorrough
walls.
9.
Themethodaccounts
for
theeffectsof
wall
roughness
on
frictional.
pressuregradients
in
transitional
flow
ã.
10.
Model
predictionsofpressuregradients
in
u
three
flow
regimeshave
been
verified
by
experimental
data
forMMHand
bentonite
~ s
flowing
through
pipes
and
annuli
withvarying
degreesof
wall
roughness.
472

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