-
9Society of PetroleumEngineers
SPE 28306
A Mechanistic Model for Cuttings TransportR.K. Clark, * Shell
Development Co., and K.L. Bickham, BET Development Co.
SPE Member
ACopyright 1994, Society of Petroleum Engineers, Inc.
Tfds paper was prepared for presenta$on at tha SPE 69th Annual
Tgchniml Conference and Exhlbltion hdd in Now ohms, LA, U. S.A.,
25+8 .Septemb.ar 1994,
This papw was SaIected for presentation by an ePE Program
Curnmittee following review of lnfomliai cantdnad In an aksfracf
silbmitted by the authm($~ contents of the paper,as presents-d,
have nof bee reviewed by the !Mety of Petroleum Engineers and am
subjwt to correction by the author(s). The material, as presented,
does not necessarily reflectany position of the Sodety of Petroleum
Engineers, its officers, or members. Papers presented af SPE
meetings are subject 10 publication re.dew by Ed[torial Committees
of the Sodetyof Petmle.m Engleers, permission to copy Is restricted
to an abstract of net ore than SCU words. IIlusttafions may not be
copied. The abstract sfm.ld wn!ah conspicuous .Mmwfedgmemof wher8
and by whom the paper Is preswtod, write Llbrarlan, SPE, P,O. Box
632s!36, Richardson, TX 7503S-3336, U .e,A, Telex, 183245
SPEUT,
ABSTRACT*
Acuttinggeneratedatthebit maybe transported to the
surfacabyseveral different mechemismsas it moves along the
wellbore. Thespecific mecbenism depends on the wellbore angle. For
highangles, where astationery cuttings bed can form, transport
isvie arolling mechanism. In intermediate angles, where a
churning,moving cuttings bed can form, transport is via a
liftingmechaniem. At near-vertical angles, particle settling
determinestransport. The model described below combinee fluid
mechanicaltreatments ofthese mecbanisme intoaforrnat for easy
analysis ofcuttings transport in wells of any configuration.
INTRODUCTION
Of the many functions that aeperformedbythe drillingfluid,
themost important ie to transport cuttinge from the bit up
theamndue to the surface. If the cuttinw cannot be removed from
thewellbore, drilling cannot proceed for long. Transport is
usuallynot a problem if the well is near verticaI. However,
considerabledifficulties caa occur when the well is being drilled
directionallyas cuttings may accumulate either in a station~ bed at
holeangles above about 500or in a moving, churning bed at lower
hole
! angles, Drilling probleme that may result include etuckpipe,
lostcirculation, high torque end drag end poor cement jobe.
Theseverity of such problems depends on the amount smdlocation
ofcuttings distributed along the wellbore.
The problem of cuttings transport in vertical wells has
beenstudied for memyyesm, with the earliest analysis of the
problembeing that of Pigott.l The transport efficiency in verticrd
wells is
...*References end Table 3 at end of paper.
usually assessed by determining the settling velocity, which
isdependent on particle sise, density and shape, the drilling
fluidrheology smdvelocity,and the hole/pipe configuration. A
recentlydeveloped correlation for settling velocity of irragubdy
shapedparticles in drilling fluids is that of Chien.2
Since the early 1980s, cuttinge traneport studies
havefocuesedoninclined wellbores, andenex~neivebody
ofliterature
on both experimental end modeling work has
developed.Experimental work on cuttin~ traneport in inclined
wellboreshaebeenconducted usingflowloopsat the University
ofl?ulsas-sand elsewhere.g- 11
Some of the more recent modeling etudies are those of Luoand
Bern,12Fordet eL,18Larsen,Pilehvari
andAser,14andRaei.16LuoandBern12and Fordet al.lspreeentmathematicel
modaIsfordeterminingt.heminimum fluidvelocityfor
trrmsportingcuttingswithout the formation of a cuttings bed. These
are physicallybased models thathavebaanvalidatedagainst
axperimentrddata,Lao and Berns model has also been compared with
field data.lqL.emens model is based on empirical correlations
derived fi-omexperimental data generated in a 35 ft long 5-in.
diame~r flowloop.14 Thie model can be used to predict a cuttings
bed height ifthe flow ie sub-critical, i.e., below the velocity
required to keepall cuttings moving upward.
Rasi assumes that a cuttings bed will form end predicts
theheight of the bed15and the open area above it. This area is
thencompsxed with the cross-sectional areaof thebit and stabilizers
tosee if they can pass through the non-bed area without
ditlkultyThemodelsofLaoandBern,12 Larsenetal,,14mdRagi15 mevddfor
hole angles greater than 50 or so where a stationary cuttingsbed
may fofi. The model of Ford et al. can be applied at any
139
-
,2 A MECHANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 28306
wellbore angle.la The models are accessible via main frame
orpersonal computers end uee readily available data as
inputparameters.
Laboratory experience indicates that the flow rate, if
highenough, will always remove the cuttings for any fluid, hole
size,and hole angle. Unfortunately, flow rates high enough
totransport cuttings up and out oftheennuluseffectively
cannotbeused in my welis due to limited pump capacity and/or
higheurfeee ordownhole dynamic pressures. Thw ieparticularly
truefor high angles with hole @eZ@ger then 12?4 in. High
rotaryspeeds and backreamin g are often used when flow rate does
notsuffice.
Drilling fluid rheology plays en important role, _althoughoften
there are exmflicting statements as to whether the mudshould be
thick or thin for effective transport. It is common whendrilling
high-angle wells for elevated low shem-rate theologies tobe
spec~led. Ty@callx the Fmm 6-rpm and 3-rpm dial readingsare set at
some level thought to aidin hole cleaning at high angles.Many
settheselowshearreadingz (inlbf/100ft2)equivalenttothehole
sizeininches. Thisrecommendationas wellazotherrulesofthumb have
been presented by Zamora and Henson.18~17
The model described below was developed to allow acomplete
cuttings transport analysis for the entire well, fromsurface to the
bit. The mechanisms which dominate withindflerent ranges of
weilboreengle areuzed to predict cuttings bedheights and ,=ukir
cuttimgs concentrations as functions ofoperating parameters (flow
rate, penetration rate), wellboreconf@ration (depth, hole angle,
hole sizeorcasinglD, pipe size),fluid properties (density,
rheology), and cuttings characteristics(density size, bed porosity
angle of repose), Parameters that arenot currently taken iriti
account include pipe eccentricity androtary speed.
This paper has three major section% (1) the fwst identifiesthe
modes of trmmport and outlines the mathematicaldevelopment of the
model, some of which is given intheappendix;(2) the- model is
compared with data generated in flow loopexperiments; and (3) three
applications of the model are used toshow its versatility in
addressing a variety of cuttings transportproblems.
CRITICAL VELOCITY MECHANISTIC MODELS
When theannulermudvelociiy ishighenough, elldrillcuttimgs inthe
wellbore aretransportedupwerd (F@re 1). Moreoveq for thegeneral
case, the annular mud velocity needs only to exeeed thecuttings bed
buildup conditions in the most sensitive spot eio-ngthe wellbore.
There, as the annular mud velocity slowly andcontinuously is
decreased, a state is reached where some cuttingsare lost from the
flow.
The notion of acritical transport condition leads to a theoryfor
czdculating the equilibrium cuttimgsbed height. First, with asteady
mud flow rate, a decrease in the wellbore annular mwaresults in a
mud velocity increase. For sub-critical mud velociiyconditions, the
theory is that the cuttings bed will build. As it
builds, the mud velocity over the bed increases. The
cuttingsbuildup process continues until the mud veloeity over the
bedssurface eventually reaches the critical value. At that
condition,the bed height remains unchanged. If additional cuttings
axedeposited onthebed, themudvelocityin theneighborhoodofthatregion
exceeds the critical velocity As aresr.dt,the stronger fluidforces
will dislodge the protruding outtinge. After these extra
cuttings are moved downstream, the local equilibrium bed
heightis then re-established. Thus, the equilibrium bed height
isformulated as a function of the critical mud veloci~.Furthermore,
the critical transport velocity is the criticalvelocity that gives
a zero cuttings bed height.
1Wellbore 7< U(Cuttings J-.nMud velocity profile(x and z
components) sFormation -%
Fig. 1 Schematic of cuttings transport in an
inclinedwellbore.
During laboratory flowlooptests,three si~]cant
patternsofcuttingsmovement were observed. They zrerolling, lifting,
andsettling; a different pattern seems to dominate the cuttings
bedformingprocessineachoftbreerangesof wellbore angles.
Athighangles, the transport pattern is rollinG namely, the cuttings
rollend bounce along the bed surface. At lower wellbore angles
wherethe wellbores complementary qngle is greater than the
cutting%angle of repose, cuttings are lifted from a churning
fluidized bed.Atnear-verticaltoverticalwellbore angles,thecuttings
sreahnostuniformly distributedthroughoutennular cross
setilonendsettledownhole against the flowing mud.
There me several mechanisms that could possibly play amajor put
in the cuttings transport process within a particularflow pattern.
In the following sections, severaI equations are
140
-
SPE 28306 R, H. CLARK AND K L. BICRHAM 3
derived for the three patterns. However, the governingmechanism
is the one which dominates the flow at a particularwellbore angle.
Two mechanisms are based on the forces requiredto displace
asingleprotrudingcuttingfrombeds surf-, namelythese equations
calculate the velocities that causes acuttingto beeither rolled or
lifted from its resting place. The third equation isbasedonthe
Kelvin-Hehnholtz stabiMyofthemudlayerflowingover the fluidizedbed.
Finally the fourth equation is baeedonthesettling velocity of the
cuttings, that is, the annular velocityreqtied to limit the
suspended cuttings concentration to fivepercent by volume in the
flowing mud stream.
Equilibrium Cuttinge Bed Height Models. Figure 2 shows a
stationary cuttingebedthathes formedonthelower wellbore wallin
an inclined well with a wellbore angle, a. At high wellboreanglee
where the wellbores complementary engleis less than
thecuttingsengle ofrepose,~, astationarycuttings bed accumulatesin
the lower part of the wellbore cross section. When the
wellborecomplemental angle, 90- a, is Iesethsm$, the outtinghae
tobeeither rolled or lifted from the bed surface in order to
move.Suppose that the cuttings bed height is in equilibrium with
theprevailing conditions. If the dynamic forces acting on
thestationmy cutting can be calculated as a fmction of local
mudvelocity, U, then the mud circulation rate needed to dislodge
thecutting can be determined. This notion is en exteneion of work
inother areas, such es sedimentation,18~Ig,zo soil erosion,zl
andslurry transport.zz
Fig. 2 Forces acting on a cutting on a cuttimgsbed.
A number of forces act, on the protruding cuttimg. Thecuttingis
assumed tobe sphericel with avoid-free interior. It hasadiameter,
d, and a material d&eit~ e.. Furth&more, it is
heldstationery byareactive force, FR. Thie force acts
throughboththepoint contact, a, at an angle, 13,end the cuttings
center of gravity.The cuttiige bed has an angle of repose,+. The
mud density is Q,
end the muds rheology is assumed to be governed by
theHerschel-BuIkley viscosity law. The static forces are
thebuoyancy force, F~ gravity force, Fg, and the plastic force,
FP,which is due to the yield stressof the mud. The dynamic forces
arethe dragforce, FD,Iiftforce, FL,and pressure gradient force,
FAPTheysxeallzssumedtoaot throughthecenterofgravity. The
mudcirculation rata is held constant.
RollingMechaniem. Forthecsaeofrolling, themomentsduetoforces are
summed around the support point, a(x,z); nmnel~
IxI(F. +FJ+ lz\@. -Fp)+g(F, -F,)=O (1)
where the length of the moment arm for the buoyancy
andgravityforces is
t = Izl(sina + cosa/ten@) (2)
Moreoveq O Othersera derived in the appendix.
141
-
-. . ...
,
4 A MECHANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 28306
The drag force,
F~ = C~~@J2, (5)
the lift force,
FL = C@ QU9, (6)
the buoyancy force,
Fb = gQ~, (7)
and the gravity force,
F. = gQ=~ (8)
where CDis the drag coefficient, CLis the lift coefficient, and
g isthe gravitational constant.
The following two equations are derived in the appendix
(Equation (A-1) end Equation (A-4)). The plastic force,
Fp = =[$ + (Jx/2 - @) Sill@ - cos@sin@], (9)
and the pressure force,
ndaF& = r~, (lo)
where
(11)
where Dhyd is the hydraulic diameter of the flow area
(sesEquation (16) and Equation (17)), P is the pressure, z~is the
wdshear stress, end ~ is the mud yield stress.
Rolling and Lifting Bed Height Equations. Two
equationsforcriticrdvelocity may beobtainedbysubstitutingEquations
(2)end (3) end the ancillary equations (Equations (5)-(11))
intoEquations (1) and (4). At high wellbore angles, one of
theseresulting equations for critical velocity may govern the flow.
Forthe case of rolling, the geverningequation for the critical
velocityis
- [.,4[3~@ + (rc/2 @) SinzI$ - COS!$Sill@)&n@ + dg(Qc
Q)(cosa + 5~ati@) drl 1
1/2
u=3Q(C~ + CLtan@)
(12)
For the csae of lifting, the governing equation for the critical
veloci~ is
.. . .
[
1/24[3~@ + (x/2 - @) Sinz@ cos @SkI@) + dg(Qc - Q)Sins]
-u = 3QCL 1 (13). . ,,- -.. .
Both Equation (12) and Equation (13) give a value for
thecritical velocity of a cutting. The velocities calculated by
theseequations are the undisturbed velocities, that is, the axial
velocityacting above the cuttings bed at a point that would be
occupied bythe cuttings center ifit were in place. These equations
calculatethe velocities that would either roll or lift the cutting
from itsresting place. In general, these two mkulated vekes will
bedifferent. In such cases, the lower value will be the dominent
oneproviding that other conditions cmemet.
Kelvin-Helmholtz Stability Model. When the behavior ofthe
cuttings is observedat low wellbore angles intheflowloop, thenature
of the mud and cuttings slurry is a churning motion. Theprocess is
reminiscent of the behavior of a gas-liquid flow whenits flow
pattern is changing from stratfled to slug flow. The
app~ce Oftie fluidized bed is similar to the liquid layer,andthe
mud layer flowingover the bed behaves like the gas layer.
Theinterface between these layers has a wavy churning
nature.Occasionally, wisps of cuttings are swept up into the mud
layer.There, they are carried downstiefi and settle back into
thefluidized bed. The process is persistmt, and it appears to
berandom.
Coneiderthestratifiedflow arrangement ehowninFigure 3.There is a
nearly cuttings-flee mud layer flowing over a mostlycuttinge-fiiled
fluid layer. A smell-amplitude wave propagates atthe inter&e as
long as the flowing conditions are stable. TheinviscidKeIvin
Helmholtz stability theory provides amethodforpredicting the onset
of unstable conditions between inertial andgravitational forces
acting on the interface,z5 that is, the value ofmud velocity that
causes the lower layer to disperse cuttingsthroughout the entire
cross section. Wallis end Dobson20 give aclear description of the
instabili@ condition for stratifiedgas-liquid flow. Their result is
adopted here as follows:
TJmk >
[
Dqg(Qb Q)sinaQ
@b = Q.(1
1/2
d( )]]
el~1w
(14)
(15)
Dq is the equivalent diameter of the area open to flow, end ~b
isthe bed porosity. Sometimes, Qbis called the submerged
bulkdensity Whentheaverage mixturevelocity, Um~, intheopenareaabove
the bed exceeds the RHS of Equation (14), the
interfacebetweenthelayers is unstable. The minimum transition
velocityis when U~ equals the RHS of Equation (14).
142
-
wSPE 28306 R. K. CLARK ANf) K L. [email protected] 5
Fig. 3 Stratified flow ofmudoverafluidized cuttingsbed.
Equations (12), (13), end (14) are theequilibriumbed
heightequations that calculate a criticef velocily condition.
Therelationship between the two different velocities, U and
Um~,needs to be emphasized here. The velocity, U, is the local
axialvelocity acting ahove the cuttings bed at a point where
thecuttings center would be if it were in place. The velocity, Uti,
isthe average flow velocity in the mea open to flow. Um~ is
easilyobtained from the operating conditions; however, the
Iocdv&city, U, is determined from fluid mechanical
relationships.
Five Peroent Mrmirnum Coneentrntion Model. Forlow-angle
conditions, Figure 4 chows a schematic of the
cuttingstransportproeess inaHerschel -Bulldey
fluidunderleminerflowconditions. The area which is open to flow is
characterized as atube insteadofanenmdus. This simplifies the
wellbore geometryThe tube diameterisbaeed onthehydraulicdiameter
forpreseuredrop calculations and on the equivalent diameter for
velocitycalculations, eo that the equatione derived in this section
can beused whether there ie a stationary cuttings bed or not.
Since drilfingmudoften exhibits ayield streee,there maybea
regio~ netw the center of the croes eeotion, where the shearstress
is less than the yield stress. There, the mud will move as aplug,
i.e., rigid body motion. The plug velocity is Up The
averagecuttings concentration and velocity in the plug are CPand
UcP,respectively. In the snnularregionaroundtheplug,
themudflowewith a velocity gradient and behaves as a viecous fluid.
Theaverage annular velocity of the mud in this region is Ua,
Inaddkion, for the cuttings in this region, theaverage
concentrationend velocity are wand U=, respectively.
Croee-Seetional Geometry First, let us define some basicwellbore
geometry. The hydraulic diemeter is defined es fourtimes the flow
ereadividedby the length of the wetted perimeter;namely,
D4 X croes-sectional area
hyd = (wetted perimeter) (16)
IT&relationship canbeusedtodetermine thehydraulicdimneterof
the area open to flow above the cuttings bed. For just thewellbore
anmdus, the hydraulic diameter of the weflbore crosesection (with
no cuttings present) is
D =D~-DP (17)
where Dh is the wellbore diameter, in., andDPis the driilpipe
OD,in. The equivalent diameter is defined as
De~ = m (18)
where A is the area open to flow. For the wellbore anrmlus,
theequivalent diameter is
% = m ()
The plug diemeter ratio is
1
cutting ~
mm
!
velocityprofile
annular ,1,region - -
0
0:
plug region - ,,,,.!
mixlure:1 .
velociiy \profile
k*
Fig. 4- Mixture and cuttings velocity profiles in aHerschel
BuMey fluid under laminar flow.
Flow Conditions. The mixture veloci~ is
(21)
where Qm is the volumetric flow rate of the mud and Q is
thevolumetric flow rate of the cuttings which depends on the bit
sizeend the penetration rate. In addition, the mixture velocity can
becalculated from the average plug end anmdus velocities in
theequivalent pipq namely,
Umk = U*(1 q + Uplf (22)
143
-
.6 A MECHANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 28306
Cuttings Concentration. The feed concentration
isdefmedasfollows:
0 &(23)
The average concentration, c, of cuttings in a short segment
withlength, Az, and cross-sectional are+ ~ can ha calculated
asfollows
c =c.(1~)+cJ:. w)
The cuttings concentrations in the plug and annular regions
areassumed equal. This means that the suspended cuttings
areuniformly distributedacroes the ereaopentoflow. Obviously,
thishas a major impact, and it probably is a function of
wellboregeometry, mud properties, cuttings properties, and
operatingconditions. It could stand alone as a research topic.
Thus, weobtain
u =_.cu.(l c). . c-c.
(25)
where
u. = U=(I ~) uq~ (26)
is the average settling velocity in the axial direction.
Thecomponents of the settling velocities (see appendiz) in the
axialdirection are
u= = Fl[c, R~, Uiz ] (27)
and
SP = Fz[U~ ,32] (28)
where
u: [ 11/2_ 4dg(QC @
3QCD (29)
[{
1[2
u~ =4 dg(e. - Q)
iccy1]
cosa, (30)~3
dUQReP = w.
(31)
3tys = dg(@c Q) (32)
CDis the drag coefficient of a sphere, ~ is the yield stress of
themud, end Vais the apparent visc6ii@ of the mud at a sheer
rateresulting from the settling cutting.
The value calculated using Equation (25) is the
minimumacceptable mixture velocity
requiredforacuttingsconcentration,c. Plgott recommended that the
concentration of suspendedcuttings be a value less than five
percent.l With this limit(C = 0.05),Equation (25) becomes
u ==-mix 0.05- 0(33)
where ~
-
. SPE 28306 R. K. CLARK* K. L. BICKHAM 7
Foralltesta,theannular cuttingeconcentrationwasallowedto reach a
steady state, cuttings injection wqs stopped, and thecuttinge were
flushed out of the anmdue end weighed. From thecuttinge weight and
density a volume percent concentration inthe annulus was
calculated. This concentration could beconverted to a cuttings bed
height knowing the cuttings bedporosity end the po~tion of the
inner pipe.
Critical Transport Prediction. Figure 5 shows the
vieuallydetermined critical flow rate, described above, as a
function ofhole angle foraxanthrmgumfluid with the properties
listed withthe figure. The hole and pipe size, penetration rate,
and cuttingssise ere also listed. The critical flow rates
determined with thepipe concentric and with the pipe eccentric are
both indicated.
FlowRste (gpm) MudVelocily((pm)
o~ A (b
S30- 9
103-
eeffle
o~# , I
0 108?2040= ev702090WelboreAngle, deg
Mud Xanthan Gum Dsnstiy: 8.3,ppgPv: 3.5 Cp PiPeDiet 2.2 in.YP:
8.0 IM1OOR2 Hole Dla .5.0 In.YZ 2.5 b/100it2 Cuffirsw 0.18 Io_,-ROP
50.0 fph
Fig. 5 Critical transport comparison.
The solid line represents the criticrd transport
conditionpredicted by the cuttings transport model. The angle r~ge
foreach mode of trensport, eettle, Iii, and roll, is indicated.
Thepredicted criticaj transport flow rate is considerably lower
thanthe visuellydetermined criticelflowrateat ee.chefthe
fouranglestested. Thedifferencebetween prediction and experiment
here isdue to the different criteria used to determine critical
conditions.Thetramsport model prediction is effectively
amininmmpreseuredrop condition. The experimental critical flow rate
is based onvisual observation and is not amenable to analytical
modeling.
Sub-Critical Prediction. Figures 6 through 9 illustrate themodel
prediction of the ahmdsr cuttings concentration in thexsnthan gum
fluid for various hole angles, pipe positions, andcuttings sizes as
a fimction of flow rate. The measured cuttingsconcentrations
tweindicatedoneach figure. The data point atthehighest flow rate
represents the vieually determined critical flowrate. The transport
model critical flow rate occurs at the sharpbreak in the elope of
the concentration versus flow rate curve,Examination of these
figures chows good agreement between the
measured data for the large cuttings (0.43-in.) and the
modelpredictions. The quantitative egreement is not so good for
thesmall cuttings, eithough qusditatively the chsnge in
cuttings
24z -23-
a~ ,* - x Measur6d Concentric 0.42v Measured Eccentr?c 0.4S
S 16 - Predkted 0.18 PFa&h3d 0.4s
~ 10-
g -8-
v4 -
2 -
0 6OW1M13374O1O31SO 2W2222402E02WFlow Rete (gpm)
Fig. 6- Cuttings transport in a 5-in. flow loop at 30.
40 Plpa Pmiticm ~~~&+ Measured concentric 0.18
25 - V Measured Eccentric 0.18
gm - x Measured Concentric 0.4sv Measur4 Ecmnt,b 0.43
Predicled 0.18~ss -;
Pmdided 0.42
j - 0+
P 5E~ !0-
5 -
w IDI 120 140 160 180 m 2ZU 240 =0 230FlowRate (gpm)
IFig. 7 Cuttings transport in a 5-in. flow loop at 50
Pipe PosSiOn s~#~+ Measured Concentric 0.1S0 Measured Eccen!ric
0.1sx Measured Concantrio 0.43v Me&sured Eccenlric MS
Predicted
01 \24 80 IL-Qla 144 100 !80 m :FlowRate (gpm)
Fig. 8 Cuttings tronsport in a 5-in. flow loop at 70.
145
-
.8 A MECHANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 28306
40* PipePositbn CultrllSize (In
+ MeasuredConcentrk0.1835- 0 Measured Eccentk 0.i8
g ~. - x Measured Concentrb 0.43v Measured E.xan!rb 0.43
Predbtd 0.18
[ 25 - Pred!!t& 0.43
g 20 -0~ 15 -
g ,00
5-
80 IW 120 140 180 130 240 220 240 =0 280FlowRate (gpm)
Fig. 9- Cuttings transport in a 5-in. flow loop at 90.
Figures 10 end 11 compexe model predictions with
experirnenti data from tests on the 8-in. flow loop With
anextended bentonite mud. Agein, the highest flowrate data
weretaken to represent ~itical conditions. These are
againconsiderably above the model prediction. The critical flow
rateprediction in the 8-in. loop is certainly more in line with
fieldexperience then that based on visual observation.
Agreementbetween experimentendmodelpredictionis quitegoodforeach
ofthe sixhole angles. Unfortunately insuflicientdatawere takenatthe
lower hole anglee to eesese the model predictions fully.
Inflow loop tests with water and other low-viscosity fluids,the
model consistently underpredicted the annular
cuttingsconcentrations at angles above 50. It appears that the
fluidrheolo~ is given more importance in the transport model than
iausually seen in high-angle flow loop experiments.6)8
28- \, q+ 20. Mea%.s - \ 20 Pred.A 35.Mea,. 30. Pred.v 4W Mess.-
- . 40- Pred.~ to
-
30 -oil -
4 -2 -
-~
o1WSM3W4WX.3 w07m8m
Flow Rate (gpm)
ig. 10- Cuttings transport at low angles in en 8-in. flow
loop.
so%-m-2A-
gr2 -
B~ 18-g; :~ 10-.s8-E
2-0 1msN3m4cOs10 sm7c08co
FlowRate (~m)
Fig. 11- Cuttings transport at high englee in en 8-in. flow
loop.
FIELD APPLICATION
The Wttings transport model, in its easy-to-use personalcomputer
format, has been applied to many different drillingsituations. A
number of these are discussed below.
DrillingLarge-Dweter Holes in Deepwater Operations.Thefwst
stringof pipe set duringdeepwater drillingoperetions isa.SO-in.or
36-in. structural pipejetted several hundred feet belowthe mud
line. The first interval drilled end cased is for either20-in. or
26-in. caaing. This interval is usually drilled with
eeawater end viscous sweeps with mud returns to the seafloor.The
large hoIe sizesmdlow-viscosity drilling fluid (eeawater)
willresult in abuildup ofcuttings in the structuraIpipe
srmuluswhichcan, if the fracture gradient at the shoe of
thestructorcd pipe islow
enough, result in loss offluid. This loss is one of several
causes forwhat iecalled shallow water flow, i.e., abreekthrough
offluid tothe sea floor mound or away horn the structural pipe.
The cuttings transport model was used to examine thisproblem end
to identify corrective action. Table 1 lists thepredicted steady
state cuttings concentration in the enmdus of a36-in. structural
pipe (34.75-in. ID) es afimction of flow rate. Thepressureat
thebaseofthe 200 ftlong, 36-in.pipegeneratedby thiscuttings-laden
fluid is also given. If the pressure imposed by thecuttings-laden
fluid exceeds the fracture pressure at the base ofthe 36-in pipe,
fluid flow to the mud line may occur. For weak,shallow sediments in
the deepwater Gulf of Mexico, the fracturegradient maybeequivelent
to only 30 or 40 psioverhydrostaticor118 to 128 psi total.
The two portions of thetable correepondta drillinga31%-in.hole
at 50 Whr with seawater end the use of a viscous sweep(density =
8.9 lbm/gaI, plastic viscosity = 9 CP,yield point =
40 lbf/100 ft2, yield stress = 15 lbf/100 ft2). The cuttings
bulkdensity is 2.05 g/cm8 and the she ie 0.25-in. The drillpipe
size is5-in. in thie example.
146
-
.SPE 28306 R. K. CLARK AND K. L. BICKHAM 9
CUTTINGS LOADING IN 36-in. STRUCTURAL PIPE
F1OW DrillwithSeawater DrillwithSweepRata(gPm) Cuttings Pres9ure
cuttings Pressure
Concentration at Shoe Concentration at Shoe(%) (psi) (%)
(psi)
750 51.0 134 21.8 llz
1000 45.0 129 15.4 107
1250 39.8 124 12.3 104
1500 35.3 120 8.6 101
1750 31.1 116 7.0 99
2000 27.2 113 4.5 97
Pipe jetted to 200 ft balow the mudline, drilling 31%-in.
hole.
The model provides guidance on drilling the 26-in.
casinginterval such that SW1OWwater flow can be minimized. It
isobvious from Table 1 that ahigh flow rata is essential, as
areperiodic viscous sweeps, to keep the pressure at the base of
thestructural pipe at a tolerable level. Drilling continuously with
asweep would be succesefid, although the total volume of
sweeprequired for drilling the 31%-in. interval may exceed the
rigmixing capability.
The cuttings concentration levels shown in Table 1
areessentially unch~ged for each of the two d@rent
operationalprocedures in common practice in deep wate~ (1) drilling
a pilothole to the 26-in. casing point and then opening to 31Ys-in.
or(2) drilling a 31%-in. hole in one pass. The sane cuttings
loadingwill eventually occur in the 36-in. ennulus whether or not a
pilothole is drilled before the final hole size is reached. If the
cuttingsfrom the pilot hole arecleaned out of the 36-in. snnulus,
they willbuild up again asthe pilot hole is opened. Itis ilso
interesting tonote that the cuttings loading is virtually
independent ofpenetration rates that a-e typical of deepwater
operations.
If~Ything,the model may underpradict the magnitude of
the cuttings btildup, se sugges~d by comparison with
theexperimental data of Ali shown in Table 2.27 Alis data
weregenerated by placinga 10-in. diameter washout, six feet in
lengthin the verticrd 5-in. flow loop at the University of Tulsa
ACarbopol solution was used as the drilling fluid.
A similar amdysiscanalsobe-conducted to examine
cuttingsbuildupinalsrge-diameterdri~ingrk~.
Theneedforhighermudviscosity, viscous sweeps, end/or additional
flow rate by boostingthe ricer can all be aasessed end operational
practice set asnecessary. Monitoring the pressure at the base of
the riser is awayof assessing how effective such practices are at
keeping the riserclean.
Table 2CUTTINGS CONCENTRATION IN A WASHOUT
FlowRate Annular EquilibriumCuttings(gpm) Velocity
Concentration(%)
(ft/min)Experimental Predicted
100 25.8 33.0 26.8
125 32.3 24.9 21.5
150 38.7 19.5 16.7
Experimental data from M (Reference No.-27).
Redevelopment Drilling. Redevelopment of axistiig fieldsoften
involvae reentering an old well, cutting a window, andrhilling out
to a newbottomhole location. Such wells czn havecompkx directional
progrmns. This was the rase in a recentoffshore well in which
awindow wascut inacurved conductor, thewell kicked to an angle of
ovar 40, droppad to near-vertical, andthen turned sharply and
eventually completed as a horizontalwefl. During drilling of the
12]/!-in. hole at an angle near 85,problems were axperiencad on
strip out of the hole at ameseureddepthof 6710 ft (5700ft TVD). It
tookexteneivebackresmingandcirculation to compIete the trip out of
the hole successftily
The output for en analysis of this situation by the
cuttingstrsmsport model is shown in Table 3. The input
parametersinclude themud type, the rheology model chosen, the
penetrationrate, the mud flow rate, the mud properties (density,
plasticviscosity,yield point, endyield stress), end the cuttings
proparties
(density diameter, bed porosity, and angle of repose).
Themeasured depth, hole angle, hole size, and pipe size complete
theinput data required for conducting the analysis. These data
areincluded in the output es indicatad in Table 3. Note that
133/5-in.easing (12.347-in. ID)hadbean setat 3010 ftmeamu-eddepth,
andthat 5-in. drill pipe end 180 ft of 8-in. drill collars were
used.
The results of the emdysis at each depth include thefollowing:
the mud velocity in the open area above the cuttingsbed, the
equivalent circrdating density (ECD), the mud
pressure(circulatingwithoutcuttings andtotaIwithcuttinge),
thecuttingsconcentration (in the circulating mud end total in the
anmdus),the areaopen to flow,andtheheight of the cuttings bed.
Figure 12depicts much of the same information but in a format that
allowsthe location of cuttings accumulations in the wellbore to be
morereadily identified.
The asterisk in the fa right-hand column of Table 3indicates
that the cuttings accumulations at this location me in amovingbed
end will avakmche down the wellbore if the pumpsareturned off
without first circulating them out of the well. Wherethere are no
asterisks (depths from 6310 to 6525 ft), a stationsg.bed three to
four inches in height is predictad.
147
-
.10 AMECI-IANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 23306
.
Well Cia.m.
RedevelopmentWell
1
E(PI
cam. FLowArecv. ma
1amMu-#el.
Drillpip Mud 12.5 ~g ROP 50.0 @h
Pv 40.0 Cp Cko. Rate 620.0 ~mWellbore v!? 17.0 Ib/looitz
Cuttisgs 0.25 in.
VZ 6.0 lb/100f12
Fig. 12- Cuttings analysis in a redevelopment well.
Several pointa can be made flomthis analysis: (1) abuildup
of cuttiige islikelv in two intervak (z) where the hole angle
ia50or less, theseouttingsarein amovingbedandcen becirculatedoutof
the well but will avalanche down the well if not circulated
outfirs~ (3) cuttings canied in a moving bed contribute to the
totalwellbore pressure (ECD); and (4) a stationary bed can exist
atangles above 50 and up hole fkom the drill collars. Table 3
andFigure 12 show the situation as it occurred. The model
inputperametera can be varied to see what action is most likely
tocorrect the situation. Increasing the flow rate to 800
gal/reinshould be sufficient to remove cuttings effectively at
angles lessthen 50, but a flow rate graatez then 1000 gal/rein
would berequired to remove the stationary bedsat
anglesgreaterthan50.
Sincethemodelisastsady state solution, it
cannotbeusedtodetermine the circulation time needed to remove
cuttings whenthey are in a moving bed. The analysis implies that
onebottoms-up time is not sufficient, but how muchlongerthan thisis
needed to remove all cuttings is unknown. Cutfmgs in astationary
bed cannot be removed by circulation alone unless themitical
flowrate isexceeded. Suchbedscenoften beremovedonlyby mechanical
action via pipe rotation and e.xiaImovement. Thework of
Raei15indicates that a stationary bed can be tolwated ifthe
cross-sectional areas of the bottomhole assembly and bit arelees
then the area available for flow. For the exemple in Table 3,this
ereais 66.8 in.2,68% of the open-hole annuhrareaat 6310 ft.
Extended-Reach Drilling. The world record extended-reachwells
drilled by Statoil in 19912s and 1992/9329have been walldocumented.
considerable hole cleaning-related problems were
experienced when drilling the 17Yz-in.interval on the C3 well
in1991. Thisintervalwae drilled from5220
fttoaftidepthof9460ftfollowingonesidetrack. Theholeanglesrangedfrom
60 to71.Based on this experience, the 17yz-kI. interval on the
nextextended-reach well, the C2, was planned and drilled with
lower
angles etartingfrom 5 andbuild@gto84. The drill pipeusedforboth
wells was 66/8-in.
The cuttinge transport model wee used to examine
the17Y2-in.interval in these two wells. The model indicates
that,
while both moving and stationery cuttings beds were presentwhile
drilling the 17%-kI. hole in each well, the extent of thestationary
bed wee far Iesain the C2 well than in the C3. Theheights of the
stationary beds are predictad to have been aboutequal in both
wells, five to six inches depending on the hole
angle,butthetotalvolome ofcuttingsinthestationarybed intheC3
wellwas over four timee the volume in the stetionmy bed in theC2
well. This reduced cuttings volume in the C2 well resulted
inIesstimeepent backreamingat ahighrotary speed, shout
theonlypractical way cuttings can be removed in a large, high-angle
hole.Each welliapredicted to have contained about the
samevolumeofcuttings in moving beds, outtinge which can be
circulated out ofthe well given eufticient circulation time.
The cuttings transport model predicts few hole cleaningproblems
in the 12%-fi. end 8%-in. intervals in both wells, eventhough these
interva.lsweredrilledatangles of80ormore. Whilesome problems were
mentioned in the StatOilpapers, they werenot of the same magnitude
se experienced in the 17Yz-in.interwd.One of the objectives of the
well path used in the C2 well was toreduee torque end drag. The
cuttings treneport model indicatesthat the@eof path eelectadforthe
C2 wall iealsobentilcial froma hole cleaning standpoint. This has
also been noted by Raei.15Thus, one of the uses of the cuttings
transport model ie to designwell paths that yield the fewest hole
cleaning problems, assumingthe path meets all of the other
objectives as well.
CONCLUSIONS
1.
2.
3.
4.
Acuttinge transport model has been presented whiohutilizesfluid
mechanical relationships developed for the variousmodes of particle
transpork aettling,lifting, and rolling. Eachtransport mechanism is
dominant within a certain range ofwellbore
smglee.Themodelpro~desameans ofanslyzingcuttings transportasa
function of operating conditions (flow rate, penetrationrate), mud
properties (denei@, rheology), well configuration(angle, hole size,
pipe size), and cuttinge properties (densitysize, angle of repose,
bed porosity).Model predictions zwein good agreement with
experimentalcuttings transport datafor flowratesbelowcritical
conditions.Predicted flow rates for cxitica.1transport, i.e., no
bedformation, are lower then those determined visually in flowloop
experiments.This versatile model. in ita PC format. has been used
toexamine several situations where poorcuttinge trsnsporthadbean
reeponsl%lefordrillingproblems. Themodelisuseful forassessing the
problems caused, for identif~ng potentialsolutions, and for
designing well paths for optimal holecleaning.
148
-
. SPE 28306 R.K. CLARKANDK. L. BICKHAM 11
ACKNOWLEDGEMENTS
We would like to thank Shell Development Company forpermission
to publish this work. We would also like to thankDr.J.J.Aser, Dc A
Pil.ehvari,Don Richison, and the studentaaudassistants at the
University of Tulsa who assisted with the flowloop experiments.
NOMENCILWPUR.E
A
c
ca
%
%
CD
CL
d
D
D4Dh
D~d
DP
DPIUg
FbFD
FgFLFPFAp
FR,
ben
Q.Q.Repu
U;ixu.
U,*
U,px
Y.z
a~.
~YY
mea open to flow
local cuttinge concentration
local cuttings concentration outside the central core of amud
with a yield stress
cuttings feed concentration
local cuttings concentration in the central core ofa mudwith a
yield stress
dreg coeflkient
lift coefficient
cutting diameter
hydraulic diameter of the wellbore immdus
equivalent diameteq see Equation (18)wellbore diameter
hydraulic diameter, see Equation (16)
drillpipe outside diameter
diameter of the central coreof a mud @th a yield stress
buoyancy force
drag forcegravity forcelift forceplastic forcepressure force
reactive force ,.consistency index -~
moment arm for the buoyan~ and gravity forcesbehavior index
-volumetric cuttings flow rate
volumetric mud flow rate
p~lcle Reynolds numberlocal veloci~ that would act at the
cuttings center in theabsence of the cuttingaverege mixture
velocity in the rweaopen to flow
average settling velocity in the axial direction
settling velocity in the area outside theplughamud witha yield
stresssettling velocity in the plug-in a mud with a yield
stresscoordinate normal to the flowing mudyield stress parameter,
Equation (22)
axial coordhate
wellbore anglewall shcxwstress
mud yield stressshear rate - -.
Yp shear rate peet a spherer pressure gradient, see Equation
(11)
e reaction force action engle
~ plug diameter ratio
Pp spparent viecosity of mud surrounding the cutting
Q mud density
Q. cutting material density
@ angle of repose
@b bed porosity
S1 METRIC CONVERSION FACTORS
CP x 1.0 * E-o3 = pfl*S
ft X 3.048 * Eol = m
fthr X 8.466667 E-05 = m/S
fvmin X 5.08 * Eo3 = IdS
gal(U.S)/min X6.309020 E-o5 = m8/s
in. X2,54 * Eo2 = m
in? X6,4516 * Eo4 = mz
lb/100 ftz X4.788026 E-01 = Pa
lbrn/geJ(U.S.) X 1.198264 E+02 = kg/m3
lbf/in.2 (psi) X6.894757 E+03 = Pa
* Conversion factor is exact.
wFEl@ICES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Pigott,R. J.S.: MudFlowinDrilling, Dtill. andProd.Pratt,,API
(1942) 91 103.
Chien,S.l?:Settling Velocity of Irregularly ShzpedPm-titles,
paper SPE 26121 (1993).
Iyoho,A.W: Drilled-Cuttim@ Transport by Non-NewtonianDrilling
Fluids Through Inclined, Eccentric hrm~ Ph.D.dissertation, U. of
Tuls~ lldea, OK (1980).
Tornre~ PH., Iyoho, A.W, and Azar, J.J.: EzperimentelStudy of
Cuttings Transport in Dwectionel Wdls~ SPEDE(Feb. 1986) 43-56.
Okrej@ S.S.endknar, J.J.: The Effects ofMud Rheology on#mnulsr
Hole Cleaning in Dwectionrd Wells, S~~E(Aug. 1886) 297308.
_@sen, T,I.: AStudy of the Critical FluidVelosityin
CuttingsTrensport~ MS thesis, U. of N@ T@% OK (1990).
Stevenik, B.C.: Design and construction of a Large-ScaleWellbore
Simulator and Investigation of Hole Size Effects
on.Cfiti~CuttingsTrensportVelocityinHighlyIncdinedWeUs~MS thesis,
U. of l?uls~ Tds% OK (1991).
Jalukar, L.S.: A Study of Hole Size Effect on CriticsI
andSubcritical DrillingFluidVelocities in Cuttings Transport
forInclined WeIlbores~ MS thesis,ll offuls% Tulsa, OK (1993).
Brown, N.E, Bern, EA., and Weaver,A.: Cleaning DeviatedHoles:
New Experimental and Theoretical Studies, paperSPE 18636 presented
at the 1989 SPE/TADC DrillingConference, New Orleans, Feb. 28MeE
3.
149
-
.12 A MECHANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 28306
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Ford, J.l!, et al.: Experimental Investigation of Dr_~edCuttings
Transport in Inclined Borehole, paper SPE 20421presented at the
1990 SPE Annual Technical Conference endExhibition, New Orleans,
Sept. 2326.
Siffermen, T.R.sndBecker,T!E.: HoieClecminginFull-ScaleIncliied
Wellbore, SPEDE (June 1992) 115120.
Luo, Y. end Bern, F!A.: Flow-Rate Predlctione for
CleaningDeviated Wells, paper IADC/SPE 23884 presented at the1992
IADC/SPE Drilling Conference, New Orleans,Feb. 1821.
Ford, J., et al.: Development of Mathematical ModelsDescribing
Drilled Cuttinge Transport in Deviated Wells,paper 93-1102
presented at the 1993 CADE/CAODC SpringDrilling Conference, Calgary
Apr. 14-16.
Lcitseri,T.I.,Pilehvari,A.A., and~ar, J.J.: Development ofaNew
Cuttin@ Trensport Model for High-hgle Wellbores
. .
Including Horizontal Wells, paper SPE 25872 presented atthe 1993
SPE Racky Moumtein Regiourd/Low PermeabilityReservoirs Symposium,
Denver, Apr. 12 14.
Rasi, M.: Hole Cleaning in Large, High-Angle Wellbores~paper
IADC/SPE 27484 presented at the 1994 IADC/SPEDrilling Conference,
Drdlaa,Feb. 1518.
Zemor% M. ,md Hanson, F!: Rules of Thumb to
ImproveHigh-AngleHole Cleaning,Pet. Eng, Intl. (Jan.
1991)4446,48,51.
Zamora,M, end Henson, F!:MoreRulesofThumb to ImproveHigh-Angle
Hole CIeaning,Pet. Eng. Zntl.(Feb. 1891) 22,M,2627.
Einstein, H.A. rindEl=Samni, E.A.: Hydrodynamic ForcesonaRough
Wrdl,Reviews ofModernPhysics (1949) 21,No. 3,520524.
E1Semni, E.A.: Hydrodynamic Forces Acting on Particlesin the
Surface of a W,resm Bed, PhD disseti-ation,U. California, Berkeley,
CA (1949).
Coleman, N.L.: A Theoretical and Experimented Study ofDrag and
Lift Forces Acting on a Sphere Resting on aHypothetical Stresmbed,
Proceedings 12th Congress of theInterrsationalAssociatwn
forHydraulicResearch, FotiCo1lins(1967) 3,185-195.
Chepil, W?S.: The Use of Evenly Spaced Hemispheres toEvaluate
Aerodynamic Forces on the Soil Surface, Trans.,American Geophysical
Union (1958) 39, No. 3,397-404.
Wicks, M.: Transport of Solids at Low Concentration inHorizontal
Pipe, iddvances in SolidLiquidFlaw in PipesandItsApplicotwn, I.
Zandi (cd.), PergamonPress, NewYork.(1967) 101-124.
Davies, T.R.H. end Samad, M.WA.~Fluid DynamicLift on aParticlqJ.
HydraulicsDiv&ion, ASCE, (1978) 104,No. HY8,11711182. -
Blevins, R.D.: Ap~lied Fluid Dvnumics Handbook. VanNostr~d
Reinhofi-Company, Ne{York (1984)
25.
26.
27.
28.
29.
30.
31.
32:
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
Milne-Thomson, L.M.: Theoretical Hydrodynamics, 4thad.,The
Macmillan Compsmy New York (1960) 404405.
Wallis, G.B. end Dobson, J.E.: The Onset of Slugging
inHorizontal Stratitied Air-Water Flow, Intl. J. MzdtiphaaeWow
(1973) 1,173-193.
M, h; The Behavior of Drilled Cuttings in WashoutSections, MS
thesis, U. Ms% Tulsa, OK (1979).
Njaerheim, A. and Tjoettq H.: New World Record inExtended-Reach
Drilling From Platform Statfjord C,paper IADC/SPE 23349 presented
at the 1992 L4DC/SPEDrilling Conference, New Orleans, Feb.
1821.
Alfsen, T.E., et al.: Pushing the Limits for Extended
ReachDrilling: New World Record tlom Platform Stut~ord C,WellC2,
paper SPE26350 presentedat the 1993 SPEAnnualTechnical Conference
and Exhibition, Houston, Oct. 3-6.
Hill, R.H,: Tha Mothemutiad Tkeo~ of Plasticity,1986 Reprint,
Oxford Urdvereity Press, New Yorlq (1950)128-160.
Perry, R.H, and Chilton, C.H.: Ckemical EngineersHandbook,
5thcd., McGraw-Hill Book Company New York,NY (1973).
Beris, A.N., et sd.: dreeping Motion of a Sphere Though aBingham
Plastic, J Fluid Mesh. (1985) 158, 219244.
Zemora, M. end Bleier, R.: Prediction of Drilling MudRheology
Using a SimpM,ed Herechel-Bulkley Model, J.PressureVesselTech,,
Trans. ASME, (Aug. 1977)88,485-490.
Seffmen, PG.: The Lift on a Small Sphere in a Slow ShearFlow, J,
Fluid Mechanics, (1965) 22, Part 2,385-400.
SefTman,PG.: Corrigendum~J. FluidMechanics, (1968) 31,Pert
3,62%
UMherr, I?H.T., Le, T.N., rmd Tiu, C.: Characterization
ofInelasticPower-Law Fluids Using Falling Sphere Da@ TheCanudian J.
Chem. Eng. (Dec. 1976) 54, 497502,
Clii, R., Grac+ J.R., sad Weber, M.E.: Bubbles, Drops,
andParticles, Academic Pres~ New York (19781. -
Beyer, WH., cd.: CRC Standard Mat~emuticol Tables, 25thEdition,
CRC Press Inc., W Palm Beach, FIorida, (1978) 143.
Benedict, R.F!: Fundamentals of Pipe Flow, John Wiley &Sons,
New York (1980).
Dodge, D.W?and Metsner, A.B.: Turbulent Flow of Non-Newtonian
Systems, A.I.Ch.E. J. (1959) 5, No. 2, 189204.
Dodge,D.WandMetzner,AB.: Errat%A,LCh.~iJ (1962)8,No. 1,143.
Govier, G.W and Asiz, K.: Tke Flow of Complex Mixtures inPipes,
van Nostrand Reinhold Compeny, New York (1972).
Torranca, B.McK.: Friction Factors for TurbulentNon-Newtonian
Fluid Flow in Circular Pipes, Tks South&can Mech. Eng. (1963)
13, No. 3, 8991.
150,.. .
-
,
SPE 28306 R. K. CLARKANDK, L, BICKHMI 13
APPENDIX
Plastic Force Acting in the Stagnfit Mud Beneath theCutting.
Acuttingsittingonthe top surface
ofacuttingsbedwill1ikelybepoeitionedinenintersticeofeeveralneighbonngcuttingeheld
stationary by the bed. The circulating drilling mud, aroundthe
upper portion of the cutting, will be liquid end flowing.
Incontrast, thedrillingmudin theintereticebeneath thecuttingwillbe
stagnant end pleeti~ assuming the mud has a yield strees,
Slip-1ine field theory provides a method to calculate
theresultant force, Fp required to Mt.acuttingfrom astagnent
layerof drilling mud. However, several simplifying assumptions
areneeded to make the calculation tractable. HWgivesamethodthatcan
beusedto calculate themesn compressive preseureand shearstress
acting on ayield surface based o.nslip-line theory.30 Sincethe
forces are axieymmetric, the region of interest can be treatedueing
a two-dimensional coordinate system. TheSpheres motionis assumed to
be incipient. The result is
F, =% -- -[Q + (n/2 ~)smz$ - eos@ sin~]. (A-1)
Force Due To Pressure Gra&ent. The differential
forceactimgin the z-direction due to a pressure gradient is
dFAp = (Pl P2)~COS2 ~df3 (A-2)
where the upstream and downstream pressure difference can
beeapressed as
PI -Pz = rd sin~. (A-3)
PI is the upstreein pressure, PZis the downstream presswe, d
iethe diameter of the sphere, 13isenenglemeaeured fromthex-axis,and
Iis defined in Equation (11). The preesure force can be foundby
integrating Equation(A-2) from Otcin/2. The result ie
Fm = Ilcds/6 (A-4)
SettlingVedocity CorrectiouFactcrs. Perry and
Chiltongiveaprocedureforcrdculatingthehinderedsettlingeffect @q.
5-224,p. 5-64),s1 They present agraphical method (Fig. 5-82, p.
5-65) fordetermining the exponent, n in Equation (A-5), as a
function ofRep Equations (A-6), (A-7), end (A-8) were chosen to fit
theirs-shaped curve within 370error.
u. = F,[c>R%, U:,] =_ UA (1 - C)n (A-5)
wheren =. e0.0811y-1.19 (A-6)
Sgn(x) y = (0.0001 + 0.865 1X1-9V3 - A-v
and . . .,,x = -1.24 hl(ReP) 4.59. (&8)
A correction for the settling velocity of the,en~elop:
thateurro~de a cutting se~tling in: mud tith a yield stress cm
beestimated se follows. The settling velocity of the particle
andenvelope system can be found horn the continuity
equatio~namely,
. .
.
= = U,(I- p)u (A-9)where ~ is the envelope-to-particle diameter
ratio. Beris al SJ.3Zcompleted a ftita difference study end found
that theenvelope-to-particle diameter ratio for material with
differentyield stresses could be determined. The following is a
curve fit oftheir resulti.
g = y;o.47. (A-1O)
After combining Equations (A-9) and (A-1O),the following
resultis obtained
u *. F2[U~ >Ys] = U$ (1 ww). (A-11}
Herschel-Bulldey lZecosity Law. For ~pical muds, it is
~gued t~t the Herschel-Bulkb?y viscosity law is a
eatiefactoryrepresentation. ZmnoraandBleier show experimentally
that thisviscosity law represents the rheologicei nature of
drilling fluideunder most steady flow conditione.33 The
Herechel-Bulkleyviscoeity law is used to express the shear stress
as follows:
z = ~Y+ khyn (A-12)
where ~ is the yield streee, kh is the consistency index,
t= dtidr is the shear m~ (YsO), and n is the behavior
index.(When T s ~ y = Oend the strtis are equal to zero. In
otherwords, the plugs interior behaves es if it were an inelastic
solidmoving at a velocity of UP)
Lift and Drag Coefficient Models. Saffman developed anemdyticel
model of the lateral forces acting on a sphere in auniform shear
flow in a Newtonian fluid.w~35Saffinens theory is
applied to tie ~ttinge trensportbyutinga%ynolds number thatis
based on the apparent viscosity of the mud surrounding thecutting;
namely,
R% = QdU/~, (A-13)
where
KB = ~Y/YP + %yp- l). (A-14)
UMherretei. present amethodto celeulatetheaverage sheerrateof a
power law fluid flowing past a sphere graphically, Thefollowing is
a fit of their r.e801k3e
p= %[+-351 A-15)where U is the velocity of the fluid relative to
the particle. If theparticle is stationary, the velocity is the
axial velocity ahove thecuttings bed at a point that would be
occupied by the cuttingscenter if it were in place.
E1Samnilg end Einstein and E1Sm@ls present resultsof the dynamic
forces due to a flowing stream acting on rocksprotmding above a
sediment bed. Their studies focused on aturbulent-water
stremnflowing over abed of rocks. This end theSaffman models are
combined as follows
151
-
.-
.
14 A MECHANISTIC MODEL FOR CUTTINGS TRANSPORT SPE 28306
CL= [ CLS=582[~~cLs2cm(A-1,)1 CL,E= 0.09 cL,~< cm
where
(A-17)
Drag Coefficient. Clitl et rd. present the best models
forcalculating the dreg coet%cient of spherical particle in
aNewtonian fluid.37
Wdlbore Geometry Model. Figure A-1 shows that the regionsof the
wellbore cross section maybe iderWzedusing ammbinationof arcs,
chords, andsegmentsofcircular m-eea.Moreover, it showsthat the
regions may have different shapes depending on theposition of the
chords defining the top end bottom surfaces of themoving zone and
the top sun%ce of the stationmy bed. Theirshape depends on whether
these top surfaces exist, and then, ifthey are below, touching, or
above the drillpipe. The boundariesthat separate these regions are
hI and hII.
~ moving cuttings zOnO -~ Ststionatyc.ttings bed
1- !-- D,-- DhFig. A-1 Wellbore cross section with a cuttings
bed.
Relationships for the arc end chord lengths smd for thesegment
ereaz can be found in any mathematical handbook (e.g.,Beyer38). The
following mathematical anaIysis leads to a set ofrelationships
based on the segment height, h, end on the circlediameter, d. The
analysis stsxts with the following basicrelationships:
.
152
B = l-~,
arc length = d cos - l(B),
chord length = d ~, end(A-18)
segment area = ~[(arc length) - B(chord length)].
Approximate Mixture Flow Model. The wellborecross-sectional
areawhichisopen to flow ischaracterized asa tubeinstead of as an
irregularly shaped channel. This decision wasmade primexily to keep
the calculations manageable at theperzonaIcomputer level. The
development ofamore physicallyaccurate flow model would be the
basis of a maior researchprogikm. Further, a more physically
accurab model should bepursued only after the approximate model is
proved inadequate.The mud rheology is calculated using the
Herschel-Bulkley
viscosity law. For both the kaninar end turbulent flow cases,
thevelocity profde end the pressure drop equations are
required.
The pressure grdlent is sum of three component; namely
dp- 1 +*I. +%If A-)Zdza
where z is the natursl coordinate in the direction tlom the
wellbottom to its top. The first term on the right is called
theaccelerational component it is negligible for this study The
nexttwo terms m-ereferred toes the elevationdange and
fictionalpressure-gradient terms, respectively.Practicallyspesking,
atlowcirculation rates the frictional term is negligible compared
withthe elevation term. However, some of the important
resultsobtained when calculating the frictional pressure-gradient
termm-e used to celcukate the cuttings concentration, namely,
thevelocities, U, U@ end Up, end the plug diameter ratio, kP,for
the
case when the flow is leminer.Since no general enrdyticfd
solution exists for a
Herschel-Bulldey fluid flowingin en eccentric enmdus with
thedrillpipe both rotating and trsnzlating axially and laterally
the
approximate fiictiond pressure gradient is calculated from
acombination of methods. The combination accounts for both
thecemplex cross-sectional geometry of the wellbore and the
natureof the non-Newtonien fluid flowing in either a Iaminar
orhubulent state. The methods are obteined from several
sources,e.g., Benadict,3gDodge and Metzneq40>41Govier and
Aziz,42andTorrance.g
Although this approach is a practical one, it leads tosituations
ofuncertainty. Forinstance, thearmularflowgeomeiryis treated as
flow in a tube with a regidsr circular cross section.The tube
diameters chosen differently depending on the purposeof the
calculation. Ifit is desired to calculate the velocity
profile,thed@neter is chosen toequsl the annulus equivalent
diametar.On the other hand, it is equal to the hydraulic diameter
if theptiposeis to predi% tie average shezwstreis actingon the
wettedper~eter~ fiother words, to predict the pressure
gradient.
-
.,SPE28306 R.K.CLARK AND K.L.BICKHAM 15
Table 3
CUTTINGS ANALYSIS IN A REDEVELOPMENT WELL
Mud Name Synthetic-BaseViscosity Law HerscheIBulkleyDrilling
Rate (ft/hr) 50.0Mud Flow Rate (galhuin) 620.0Fluid Density
(ibm/gal) 12.5Pv (Cp) 40.0YP (lbf/100 ftz) 17.0YZ (lbf/100 ftz)
6.0Cuttings Density (g/cm3) 2.30Cuttings Diameter (in.) 0.25Bed
Porosity (%) 37.0Cuttings Angle of Repose (deg) 40.0
Program Reeuke
Survey Meas. Hole Hole Pipe Mud ECD Pressure cuttings Flow
BedPoint Depth Ang. Diem. OD Vel. Circ. Total Circ. TotiJ lwea
Ht.
(ft) (deg) (ii.) (ii.) (fpm) (Ppg) (psi) (psi) % % % (in.)
1 02 915
3 1575
4 1660
5 2165
6 2915
7 30108 Soil
9 3195
10 3750
11 4320
12 4560
13 4875
14 525015 55543
16 5700
17 5865
18 6010
19 6105
20 6245
21 6275
22 6310
23 6360
24 6435
25 6525
26 6526
27 6610
28 6709
29 6710
0.027.5
38.643.3
44,0
35.9
33.533.5
32.2
25.1
15.9
12.0
6.0
2.2
8.9
20.4
33.844.9
48.4
47.1
50.052.757.262.070.070.080.784.384.3
12.347
12.347
12.347
12.347
12.347
12.347
12.84712.25012.250
12.250
12.250
12.250
12.250
i2.250
12.250
12.250
12.250
12.250
12.250
i2.250
12.250
12.250
12.250
12.250
12.250
12.250
12.250
12.250
12.250
5.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0005.0008.0008.0008.0008.000
120128
144
147
148
139
136137136
1.23
123
123
123
123
123
123
138
149
154
154
154
176
174
170
163
178
178
178
178
12.5
12.7
12.9
12.9
13.0
13.2
13.213.213.2
13.2
13s
13.1
13.1
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.0
13.1
13.1
0 08 606
15 1002
18 115822 1315
31 1698
32 175232 175234 1858
40 2180
46 2522
48 2675
51 2679
54 312657 3325
59 8423
60 3525
62 360863 3656
65 3723
65 373766 3753
86 3774
68 3801
69 3830
69 383171 3851
73 3864
73 3864
0.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.90.80.80.80.8
0.94.7
11.0
12.2
12.4
9.4
8.07.86.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
7.811.3
13.7
13.7
13.7
19.8
19.1
18.1
16.3
0.8
0.8
0.8
0.8
69
93
82
81
80
85
878869
99
99
9999
99
99
98
88
81
78
78
76
68
70
71
74
99
99
99
99
01.2*
2.4*
2.6*
2.6*
2.1*
1.9*
1.8*1.6*
o
0000001.8
2.5*
2.8*
2.@
2.8*
3.7
3.6
3.5
3.2
0
0
0
0
*Cuttinge bed may avelanche when circulation stops if hole angle
is less than 50 degrees.
I 153
-
..,:. .,
. ..
-.
_.
