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STATOR LIFE OF A POSITIVE DISPLACEMENTDOWN-HOLE DRILLING MOTOR

R&M Energy SystemsA Unit of Robbins & Myers, Inc.

Conroe, Texas

ABSTRACTThe power section of a positive displacement drillmotor (PDM) consists of a steel rotor and a tubewith a molded elastomeric lining (stator). Powersection failures are typically due to the failure of thestator elastomer. Stator life depends on manyfactors such as design, materials of construction,and down hole operating conditions. This paperfocuses on the stator failure mechanisms and factorsaffecting stator life. An analytical method forpredicting the effect of various design and operatingparameters on the strain state and heat build-upwithin elastomers is discussed.

The effect of parameters such as rotor/stator design,down hole temperature, drilling fluid, statorelastomer properties, motor speed, and motordifferential pressure on the stator life is discussed.Non-linear finite element analysis is used toperform thermal and structural analysis on the statorelastomer. Data from laboratory accelerated lifetests on power section stators is presented todemonstrate the effect of operating conditions onstator life.

NOMENCLATURE

F loading frequency [Hz]G’ elastic modulus [psi]H hysteresis heat [BTU/hr-ft3]N number of rotor lobes∆P differential pressure across the

power section [dpsi]Q flow rate [gpm]S∆P slip or blow-by of fluid past seal lines.

A function of differential pressure acrossadjacent cavities. [number between 0 and 1]

T torque [ft-lb]Vc cavity volume; stator pitch x pumping

area [in3]W rotor speed [rpm]ε strain [in/in]tan δ ratio of viscous to elastic modulus

BACKGROUNDMud Motor Power SectionThe power section of a positive displacement drillmotor (PDM) converts the hydraulic energy of highpressure drilling fluid to mechanical energy in theform of torque output for the drill bit. A powersection consists of a helical-shaped rotor and stator.The rotor is typically made of steel and is eitherchrome plated or coated for wear resistance. The

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stator is a heat-treated steel tube lined with ahelical-shaped elastomeric insert. Figure 1 is across-sectional view of a typical power section.

LOBE

CAVITIES

Figure 1. Cross-Sectional View of a 4:5 LobePower Section.

As shown in Figure 2, the rotors have one less lobethan the stators and when the two are assembled, aseries of cavities is formed along the helical curveof the power section. Each of the cavities is sealedfrom adjacent cavities by seal lines. Seal lines areformed along the contact line between the rotor andstator and are critical to power section performanceas will be discussed later.

ROTORS

STATORS

Figure 2. Various Lobe Configurations.

The centerline of the rotor is offset from the centerof the stator by a fixed value known as the“eccentricity” of the power section. When the rotorturns inside the stator, its center moves in a circularmotion about the center of the stator. Rotation ofthe rotor about its own axis occurs simultaneouslybut it is opposite to the rotation of the rotor centerabout the stator center. Figures 3 and 4 illustrate

two positions of a power section rotor within itscorresponding stator.

Figure 3. Rotor with Lobe “A” Fully Insertedin Stator Lobe.

Figure 4. Rotor Position Rotated Approximately20 Degrees from Position in Figure 3.

During drilling operations, high pressure fluid ispumped into the top end of the power section whereit fills the first set of open cavities. The pressuredifferential across two adjacent cavities forces therotor to turn and as this occurs, adjacent cavities areopened allowing the fluid to flow progressivelydown the length of the power section. Opening andclosing of the cavities occur in a continuous,pulsationless manner causing the rotor to rotate at aspeed that is proportional to drilling fluid flow rate(Equation 1). This action converts fluid hydraulicenergy into mechanical energy. As shown inEquation 2, the torque of a power section isproportional to cavity volume and differentialpressure across the power section.

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W = [231*Q/ (N * Vc)]*S∆P (1)

T = (N*Vc*∆P)/24π (2)

Cavity volume is purely a function of power sectiondesign. As shown above, it is defined as pumping(cavity cross sectional) area multiplied by statorpitch. Moineau theory defines the maximumpumping area that can be obtained within a givenstator tube diameter. Power section speed isinversely proportional to stator pitch length. Figure5 illustrates the effect of pitch length on rotor speedat a given fluid flow rate.

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5

NORMALIZED STATOR PITCH

NO

RM

AL

IZE

D

RO

TO

R S

PE

ED

4:5 LOBE

Figure 5. Reduction in Rotor Speed withIncreasing Stator Pitch.

Pressure Rating and SlipThe recommended differential pressure of a powersection is the summation of the pressure ratings foreach individual stage. Although the definition of astage is somewhat arbitrary, it is typically defined asone pitch length of the stator. The pressuredifferential rating for an individual stage generallyranges from 100 to 300 dpsi and depends on numberof lobes, pitch length, compression fit, andelastomer physical properties. For a power section,at otherwise identical conditions, higher pressureper stage usually means lower stator life. This willbe discussed later.

The pressure rating is the differential pressure atwhich a power section should operate to achieveoptimum stator life. However, it is not uncommonduring aggressive drilling to run power sectionswell above the maximum pressure rating. In manycases users will target operation at differentialpressures just below stalling conditions. Thispractice does result in significant reduction of statorlife.

Slip is caused when high pressure fluid blows byrotor and stator seal lines. Slip results in powersection speed reduction and is defined as the percentreduction in rotor speed below maximum theoreticalfor a given flowrate. The following tablesummarizes the impact of different design andoperating parameters on power section slip.

Table I. Parameters Affecting Slip.

Parameter Effect on SlipPressure differential increase IncreaseCompression fit increase DecreaseRubber modulus increase DecreaseFlow rate increase No changeRotor/Stator wear IncreaseStator expansion due totemperature or chemical swell

Decrease

During drilling, differential pressure and slipincrease as the load on the bit increases. This causesthe rotor speed to slow down until at some pointabove maximum rated pressure, the power sectionstalls. Once the motor is stalled, all drilling fluidblows by the seal lines. The differential pressure atwhich stall is reached can be increased byincreasing compression fit between the rotor andstator. Figure 6 shows the impact of a large fitvariation on power section speed and torque output.If the rotor-stator fit becomes too tight, stator lifewill be significantly reduced. Optimal fit provides aslip efficiency that is a compromise of stall marginat maximum rated pressure and stator life.

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

NORMALIZED DIFFERENTIAL PRESSURE

NO

RM

AL

IZE

D R

OT

OR

SP

EE

D

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NO

RM

AL

IZE

D T

OR

QU

E

TIGHT

LOOSE

SPEED

TORQUE

Figure 6. Effect of Fit on Power SectionPerformance.

Figures 7 and 8 are performance charts for a typicalpower section. As the load on bit is increased, thedifferential pressure across the power section andtorque output increase while the rotor speeddecreases. The full load curve represents themaximum recommended differential pressure atwhich the power section should be operated. Notethat the pressure rating decreases as flow rate androtor speed increase. The reason for derating apower section is to achieve longer life. This will beexplained in more detail later.

0

50

100

150

200

250

0 100 200 300 400 500

DIFFERENTIAL PRESSURE (PSI)

RO

TO

R S

PE

ED

(

RP

M)

INCREASING FLOWRATE

MAX ∆P LINE

Figure 7. Typical Power Section Performance.

050

100150200250300350400450

0 100 200 300 400 500

DIFFERENTIAL PRESSURE (PSI)

TO

RQ

UE

(F

T-L

B) EFFECTIVE ∆P

OFF-BOTTOMPRESSURE LOSSES

INCREASING FLOWRATE

Figure 8. Typical Power Section Performance.

Figure 8 shows that torque output of a powersection increases essentially linearly with increasingdifferential pressure across a power section. Thepressure losses shown are the combined effects offlow losses in the entrance region of the powersection and of frictional losses between the rotorand stator. The losses are quantified as thedifferential pressure required to start the rotorturning and are dependent on drilling fluid flowrate.In the example above, the losses range from 50 psiat the lowest flowrate to 120 psi at the highestflowrate. The differential pressure needed for start-up does not contribute to torque generation by thepower section. For example, if a power section isoperated at 400 psi differential pressure and thestart-up differential pressure is 100 psi, thedifferential pressure that is effectively generatingpower is 300 psi.

FAILURE MECHANISMSOne of the most challenging aspects of utilizingpower sections for drilling operations isunderstanding and predicting failure. Power sectionfailures are primarily due to destruction of the statorelastomer. Rotor failures due to wear or chemicalattack are rare compared to stator failures and arenot discussed in this paper. Elastomer failures maybe classified as those which result in a reduction inperformance and those which are catastrophic. In

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many cases continued operation under conditions ofreduced performance will lead to catastrophicfailure. Each type of failure may be caused by avariety of reasons. In the following sections, keystator failure mechanisms and the factors thatinfluence them are categorized.

Mechanical and FatigueMechanical failure of the stator elastomer occurswhen the elastomer is overloaded beyond its stressand strain limits. Any number of the followingfactors may contribute to premature statormechanical failures: 1) excessive pressure duringaggressive drilling operations; 2) repeated stalling;or 3) high compression fit between rotor and stator.Each of these factors results in overstrain of thestator lobes beyond their mechanical limits. Figure9 is an illustration of a stator that failed under highmechanical loading.In some cases, power section stators can fail due tofatigue at mechanical loading conditions well belowthe rubber tear strength.

Figure 9. Chunked Stator Due to Overpressure.

Fatigue failures are the result of high cyclic loadingon the stator elastomer due to rotor speed. Equation3 defines the loading frequency for a power sectionstator.

F = (RPM/60)* N (3)

The cyclic loading simply defines the number oftimes a stator lobe is flexed in a unit of time. As the

number of power section lobes increases, fatiguelife decreases because the loading frequencyincreases. One method for compensating for this isto reduce rotor speed. At high loading frequencies,the strain and strain rates on the elastomer will besufficient to promote the initiation and propagationof microscopic cracks in the stator lobes. Thisphenomenon, known as fatigue crack growth,occurs under high strain and strain rate dependingon the elastomer tear strength and strain energyrelease rate. If the elastomer is subjected to strainbelow the critical level, the onset of fatigue crackgrowth may not occur even at very high frequencies.However, at strains above the critical level, crackswill initiate in the high strain region, usually in thebottom of the stator lobes, and the crack growth ratewill depend on the cyclic rate of loading. Figure 10illustrates failure of a stator operated above thecritical strain level for a given loading frequency.

Figure 10. Failed Stator Due to FatigueCrack Growth.

Thermal and Hysteresis FailuresThermal failures occur when stator elastomertemperature exceeds its rated temperature for aprolonged duration. Stator elastomer physicalproperties usually weaken as temperature increases.The weakening of the elastomer properties results inshortened stator life. High elastomer temperatures

FATIGUECRACKS ATBOTTOMOF LOBES

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may be due to down-hole temperature, hysteresisheat, or the combination of both.

Exposure to the down-hole temperature will causethe stator elastomer to expand which tightenscompression fit. Degradation of elastomer physicalproperties will occur if the down-hole temperatureis above the temperature rating of the elastomer.

Hysteresis heat generation is due to repeated flexingof the stator lobes by the rotor and the pressurizedfluid. Because elastomers are visco-elasticmaterials, a portion of the flexing energy isconverted into thermal energy. Equation 4 fromReference 1 can be used to estimate hysteresis heatgeneration within elastomers.

H = 2100* G’ * tan δ *ε2 * F (4)

The location of peak hysteresis heat build-up is nearthe center of the stator lobes. The strain in thisregion combined with the low thermal conductivityof elastomers result in this heat build-up. Figure 11shows the temperature distribution within a typicalstator due to hysteresis heat build-up. Note the 30degree F temperature build-up due to hysteresis.The heat build-up increases as power section speed,pressure differential, or compression fit is increased.The maximum temperature within the stator mayexceed the elastomer’s temperature rating, even ifthe down-hole temperature is well within theoperating limits of the stator. Therefore, at elevateddown-hole temperatures, power section life may beprolonged if the power sections are operated at slowspeed or low differential pressure.

Figure 11. Hysteresis Heat Build-up WithinStator Elastomer.

In all the cases described above, the result ofelastomer temperature exceeding its temperaturerating is: 1) the reduction of elastomer physicalproperties; and 2) the expansion of the elastomerwhich tightens rotor/stator compression fit. Thecombined thermal and mechanical effectssignificantly reduce stator life. Using an oversizestator is one method for compensating for increasedfit due to elastomer expansion.

Chemicals and AromaticsDrilling fluids are composed of many differentchemicals and are uniquely designed to improvedrilling penetration rate, prevent formation damage,allow easy clean-up, and facilitate other drillingrequirements. Some of the chemicals, syntheticoils, or aromatics used in drilling fluids weaken therubber molecular chain resulting in reduction inrubber physical properties and shrinkage or swell.Weakening of the rubber combined with a change incompression fit due to shrinkage or swell willaccelerate stator failure. Figure 12 shows an

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example of change in elastomer properties whenexposed to a common drilling fluid at 4000 psi and300 degrees F. Discussions related to the elastomercompatibility with various drilling fluids is outsidethe scope of this paper.

4000 psi at 300 Deg F

0102030405060

DrillingFluid 1

DrillingFluid 2

% L

OS

S IN

PH

YS

ICA

L

PR

OP

ER

TIE

S

Nitrile

HNBR

Figure 12. Effect of Drilling Fluid onElastomer Properties.

STATOR LIFE OPTIMIZATIONStator life is critical to all drilling operations. Inorder to achieve optimum life, stators must bedesigned and operated with knowledge of thefactors that influence life. The following sectiondescribes these factors and how each is accountedfor in design practices to optimize power sectionlife.

Rotor/Stator Interference FitInterference (compression) fit is probably the mostcritical factor that determines stator life. Optimumfit provides a balance between frictional losses,power section efficiency, and stator life. If theinterference is higher than optimal, power sectionefficiency increases because of reduced fluid slipbetween cavities (see Figure 6). At highinterference, frictional losses and rubber strainincrease dramatically, and stator life is degraded dueto high strain conditions. Laboratory tests show thatstator life is significantly reduced if compression fit

is “too loose” or “too tight.” Figure 13 shows thenormalized data obtained in the lab.

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0.8

1

0 0.5 1 1.5 2

NORMALIZED OPTIMUM COMPRESSION FIT

NO

RM

AL

IZE

D

ST

AT

OR

LIF

E

Figure 13. Stator Life is Reduced if CompressionFit is “Too Loose” or “Too Tight.”

In cases where the interference is lower thanoptimal, power section efficiency drops due toslippage of high pressure fluid between cavities andstator life decreases due to increased susceptibilityto stalls and stator wear.

The power section design process involves selectinga compression fit that will provide optimal statorlife at a specific down-hole temperature. Fitselection is made based on test data, field data, andexperience. Design parameters such as lobeconfiguration, stator pitch, and elastomer type arealso considered in the fit selection process.

After the power section design process has beencompleted and rotors and stators have beenmanufactured, proper rotor/stator fit must beselected depending on the drilling conditions.Power section manufacturers offer various rotor andstator sizes to accommodate fit selection fordifferent applications. For example, a standardrotor and stator may be used at a circulatingtemperature of 150 degrees F while a standard rotorand an oversize (OS) stator is used to achieve thesame performance and stator life at 250 degrees F.

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Oversize stators are also utilized when using drillingfluids that are known to cause elastomer swell.

Accurate measurements of rotor and stator sizes areimportant in power section fit selection. Variationsin stator sizes of as little as 0.005-0.010 inches canresult in significant changes in performance andstator life. Accurate measurement of stator profilesize and shape is extremely difficult because: 1)there are size changes with variations in ambienttemperature and humidity; 2) the internal geometryof the stator is complex; 3) the elastomer flexesduring measurement; and 4) measurementtechniques vary. Most manufacturers of powersections provide rotor and stator dimensions so thatthe operators can match the rotor and stator toachieve the desired fit for the particular application.

Operating ConditionsRunning a power section at or below maximumrecommended pressure is the primary operationalconsideration that must be made to maximize statorlife. Excessive differential pressure during drillingcauses extreme deformation of the stator lobesresulting in premature mechanical failures.

Consideration must also be made during drillingoperations for rotor speed. As shown in Figure 7,the differential pressure rating for a power sectiondecreases as rotor speed (flow rate) increases. Thereason power section pressure differential isderated with increasing rotor speed is to offset theeffect of increased rubber strain rates. If themaximum pressure rating is not derated at highrotor speed, stator life will be reduced.

ElastomersPower section stators are commonly made withnitriles (NBR) because of their excellent physicalproperties and oil resistance. Nitrile rubbers (NBR)are manufactured by copolymerization of butadienewith acrylonitrile (ACN). Typical stator rubbercompound consists of a nitrile base polymer,reinforcing materials, curatives, accelerators, and

plasticizers. Rubber compound formulations areproprietary to power section manufacturers and aredesigned to address different applications.

The majority of stator elastomer properties aredetermined by the base polymer used in thecompound. All nitrile polymers are prepared withvarying ratios of ACN. The amount of oil andsolvent resistance is based on the ACN content ofthe polymer. Compounds with 25 to 35 percentACN content are “medium”, and compounds with35 to 50 percent ACN are known as “high” ACNcompounds.

Hydrogenated nitriles (HNBR) are produced byintroducing hydrogen to dissolved nitrile elastomersto improve its physical properties. The HNBRproperties that are most relevant to power sectionstators are high tensile strength, high modulusretention at elevated temperatures, high hot tearresistance, improved oil and solvent resistance overNBRs, and heat resistance. The hydrogenation levelof an HNBR varies from 80 to 99 percent. HNBRswith 90 percent or higher hydrogenation aresometimes referred to as Highly Saturated Nitrilesor HSN.

A stator rubber compound is designed for differentdrilling applications. Typically, HSNs are used forhigh temperature applications and high ACNcompounds are used for applications with morearomatic oil-based drilling fluids. Compound designwill determine rubber properties such as tensilestrength, hysteresis heat build-up, fatigue life, andmodulus retention all of which are critical to apower section’s operation and life.

ANALYTICAL MODELLINGThe following section describes a method forpredicting stator life under various operationalconditions. The results may be used as a guidelineto maximize stator life.

Analytical technique for stator life prediction

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A non-linear finite element analysis (FEA) approachcan be used to predict the elastomer strain levels ofa typical power section with various interferencefits, at different operational and down-holeconditions, and for different rotor positions withinthe stator. The calculated strain state can then beutilized as input for predicting hysteresis heat build-up within the elastomer. Earlier work (Delpassand,1995) describes the two-part analysis used tocalculate heat build-up within stator elastomers.

Figure 14. Life Prediction Analysis Flow Chart.

Next, empirical data may be employed to determinethe physical property reduction of an elastomer astemperature increases. Finally, an estimate of statorlife can be made based on the stress and strain of theelastomer at the temperature generated within thecenter of the stator lobes.

At the strain levels encountered in the statorelastomer at down-hole conditions, the materialproperties are non-linear. This is shown inDelpassand (1995). Therefore, use of non-linearelastomer properties as determined throughlaboratory testing is important in order to achieveaccurate results.

To conduct FEA, geometrical and thermal boundaryconditions must be simulated. The structuralboundary conditions imposed on the statorelastomer are compression fit between the rotor andstator, hydraulic pressure across the stator lobesfrom the drilling fluid, and elastomer-to-tube bond.Radial forces caused by the eccentric motion of therotor can be ignored for smaller power sections.

The thermal boundary conditions on the stator areforced convection between the drilling fluid and theinternal surfaces of the stator and the tube outsidewall. Hysteresis heat input to the elastomer iscalculated using Equation 4.

ResultsThe following section provides an example of thestator life prediction method described above.Table II lists the selected operating conditions forthe analysis.

Table II. Example Operating Conditions.

PARAMETERElastomer Typical NitrileAmbient Compression Fit [in] 0.010Circulating Temp [degrees F] 150Rotor Speed [rpm] 250Differential Pressure perStage [psi]

125

Figure 15 illustrates the predicted strain state in theelastomer of a 5-lobe stator at the down-holeoperating conditions in Table II and with the rotorin the top dead center (TDC) position. For thepurpose of analysis, the rotor position whichresulted in the highest strain levels was utilized inheat generation predictions.

OperatingConditions

DesignParameters

ElastomerStrain andStrain Rate

HysteresisHeat Build-

up

MechanicalStrain and

Stress

EmpiricalLife

Prediction

LaboratoryData

Iterations

Iterations

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Figure 15. Strain State with Rotor in TDCPosition. Lighter Sections ShowHigher Strain.

Figure 16. Elastomer Deflection with RotorPosition 15degrees from TDC.

Figure 16 shows the rubber deflection and strain atthe above conditions but with the rotor positioned15 degrees from TDC.

Figure 11 illustrates the predicted temperaturedistribution within the stator elastomer at theselected conditions. In the example given, the

temperature within the center of the stator lobes is30 degrees higher than the circulating temperature.

The foregoing figures illustrate the effect of designand operating conditions on the heat generationwithin the elastomer of a stator. Figure 17 showsthe strain-energy capability of a typical nitrile as afunction of elastomer temperature. Strain energy isdefined as the area under the rubber stress-straincurve.

0

100

200

300

400

500

150 200 250 300

TEMPERATURE (F)

ST

RA

IN E

NE

RG

Y (

KP

SI)

Nitrile

HNBR

Figure 17. Elastomer Strain Energy Capability.

Finally, knowledge of elastomer strain energyreduction due to temperature can be correlated tostator life. Table III shows an example of stator lifeprediction data for a 6.75” diameter 4:5 lobe powersection. In the cases selected, FEA was used topredict the rubber strain and temperature build-up atvarious circulating temperatures, pressures perstage, and rotor speeds. The predicted maximumstator temperature was then used in conjunctionwith Figure 17 to determine the rubber strainenergy. Finally, the results were correlated withstator life test data collected under the first set ofconditions in Table III. The analysis does notinclude the impact of drilling fluid compatibility orany other specific operating conditions.

High strainregion wherefatigue crackstypically form.

Elastomerdeflection dueto compression.

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Table III. Life Prediction For 6.75” 4:5 Lobe.

Circulating Temperature

(F)

Pressure per Stage

(PSI)

Rotor Speed (RPM)

Maximum Stator

Temp (F)

Normalized Rubber Strain

Rubber Strain

Energy (KPSI)

Normalized Stator Life

Stator Life Estimates

(hours)150 100 400 238 1 125 1 500200 200 400 344 2.3 49 0.17 87200 100 600 332 1.2 52 0.33 167250 100 400 338 1.3 51 0.31 154

The figures 18 and 19 illustrate a few of the testresults recently obtained in the laboratory. Thefigures show the effect of rotor speed anddifferential pressure on heat build-up within theelastomer.

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80

90

100

110

120

0 200 400 600 800 1000ROTOR SPEED (RPM)

EL

AS

TO

ME

R

TE

MP

(F

)

Figure 18. Heat Generation Due to Rotor Speed.

7:8 LOBE, 450 GPM

0

20

40

60

80

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120

0 100 200 300 400 500 600 700 800

DIFFERENTIAL PRESSURE (PSI)

ST

AT

OR

TE

MP

IN

CR

EA

SE

(F

)

TEST DATA

Figure 19. Heat Generation Due toDifferential Pressure.

CONCLUSIONSPower section stators typically fail due to highmechanical loading, fatigue, drilling fluid

incompatibility, or high temperature. Mechanicalfailures occur when the stator elastomer isoverloaded beyond its stress and strain levels.Excessive pressures, repeated stalls, or too muchcompression between rotor and stator result in amechanical failure. Fatigue failures occur whenelastomer strains are above critical limits and thestator lobes are subject to high cyclic loading.Cracks due to fatigue are often initiated in thetransition between the crests and valleys of thestator lobes and lead to stator failure. Some of thechemicals and oils used in drilling fluids change thephysical properties of the stator elastomers.Weakening of the rubber combined with a change inthe compression fit due to shrinkage or swell willaccelerate stator failure. High temperature is one ofthe most important parameters leading to a powersection stator failure. High elastomer temperaturesare due to down-hole conditions, hysteresis heatbuild-up, or the combination of both. At elevatedtemperatures, elastomer properties are degraded andall failure modes are accelerated.

In order to maximize stator life, compression fitbetween the rotor and stator must be selected for thedown-hole conditions. In addition, power sectiondifferential pressure should be reduced as rotorspeed is increased to maintain stator life. Finally,the stator elastomer must be carefully selected toinsure compatibility with the drilling fluid.

REFERENCESDelpassand, Majid, 1995, “Mud Motor StatorTemperature Analysis Technique”, ASME DrillingTechnology, Book No. H00920.

200 psi

100 psi

50 psi

0 psi

4:5 LOBE