research article mathematical modeling applied to drilling...

Research ArticleMathematical Modeling Applied to DrillingEngineering An Application of Bourgoyne and Young ROPModel to a Presalt Case Study

Andreas Nascimento1 David Tamas Kutas2 Asad Elmgerbi2

Gerhard Thonhauser2 and Mauro Hugo Mathias1

1Universidade Estadual Paulista (UNESP) Faculdade de Engenharia Campus de Guaratingueta (FEG)Departamento de Mecanica (DME)PRH48-ANP Avenida Ariberto Pereira da Cunha 333Portal das Colinas 12516-410 Guaratingueta SP Brazil2Montanuniversitat Leoben (MUL) Department of Petroleum Engineering (DPE) Chair of Drilling andCompletion Engineering (CDC) Erzherzog-Johann-Straszlige 3 8700 Leoben Austria

Correspondence should be addressed to Andreas Nascimento andreasnascimentogmailcom

Received 25 June 2015 Revised 18 August 2015 Accepted 20 August 2015

Academic Editor Reza Jazar

Copyright copy 2015 Andreas Nascimento et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Several mathematical ROPmodels were developed in the last five decades in the petroleum industry departing from rather simplebut less reliable R-W-N (drilling rate weight on bit and rotary speed) formulations until the arrival to more comprehensive andcomplete approaches such as the Bourgoyne and Young ROP model (BYM) widely used in the petroleum industry The paperemphasizes the BYM formulation how it is applied in terms of ROP modeling identifies the main drilling parameters drivingeach subfunction and introduces how they were developed the paper is also addressing the normalization factors and modelingcoefficients which have significant influence on the model The present work details three simulations aiming to understand theapproach by applying the formulation in a presalt layer and how some modification of the main method may impact the modelingof the fitting process The simulation runs show that the relative error measures can be seen as the most reliable fitting verificationon top of R-squared Applying normalization factors and by allowing a more wide range of applicable drillability coefficients theregression could allow better fitting of the simulation to real data from 54 to 73 which is an improvement of about 20

1 Introduction

Drilling parameters such as weight on bit (WOB) and rotaryspeed (RPM) while performing the drilling activity itself aresometimes based on normal field operation practices ratherthan calculated for correct optimized values If these weightsand speeds vary from the optimized values significantdrilling related costs increase may be expected and there-fore optimization can lead to significant capital expenditurereduction It is especially relevant in case of offshore drillingoperations where rigs spend much less of total drilling timeon locationwith actual drilling function compared to onshoreoperations

Several drilling models were proposed in the past toexplain the effects of drilling parameters environment andgeology effect on the ROP Significant research started in theend of the first half of the 20th Century and the first modelswere based on R-W-N (ROP WOB and RPM) equationsmainly driven by empirical exponents multiplied by propor-tionality constants to take influencing variables into accountbut later laboratory tests revealed that R-W-N equationsshowed reliable results only in case of perfect hole cleaningconditions The evolution of these modeling started basicallyin 1960 with Cunningham (1960) followed by a chain ofchanges and further researches addressed by Maurer (1962)Galle and Woods (1964) Bingham (1965) Bourgoyne Jr and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 631290 9 pageshttpdxdoiorg1011552015631290

2 Mathematical Problems in Engineering

Young Jr (1974) and Warren (1981) actual and subsequentmodels were mainly based on further improvements on topof just-mentioned root ones [1ndash6]

Apart of the BYM ROPmodel detailed in the subsequentchapters the next equations below by Galle and Woods

(1964) model (1) andWarren (1981) model (2) are shown as amodel comparison reference and not as the main focus of thepaper

ROP = 119862119891 sdotWOBsf

119896sdot [119890

(minus100RPMsf2)sdot RPMsf

1198871+ 1198872 sdot RPMsf (1 minus 119890

(minus100RPMsf2))]

(0928125 sdotOD2bit + 6 sdotODbit + 1)119901

(1)

ROP = 119870 sdotRPMsf sdot (WOBsf minus (WOBsf)119905)

2

OD2bit sdot 1198782 (2)

where 1198871 is 075 or 0428 for soft and hard formationsrespectively [unitless] 1198872 is 05 or 02 for soft and hardformations respectively [unitless] 119901 is tooth wear coefficient[unitless] 119896 is coefficient accounting for WOB influence onROP [unitless] 119862119891 is coefficient accounting for bit typehydraulics drilling fluid and formation strength [unitless]and 119870 is constant dependent on drill-bit dullness and for-mation abrasiveness as a consequence of drilling conditions[unitless]

Research and development continued yet one of themostcomprehensive ROP models is the one proposed by Bour-goyne Jr and Young Jr published in 1974 which considersthe effects of the depth the characteristics of the formationbeing drilled the drill-bit size the mechanical factors ofthe drilling process (ie WOB and RPM) and the mudsystem properties allowing each one to be adjusted by fittingcoefficients [4 7]

2 BYM ROP Model

The modelrsquos main equation (3) consists of 8 subfunctionswhich act and have significant influence on the ROP perfor-mance [2]

ROP = 1198911 lowast 1198912 lowast 1198913 lowast 1198914 lowast 1198915 lowast 1198916 lowast 1198917 lowast 1198918 (3)

where 1198911 is effect of formation strength 1198912 is effect of depthand compaction 1198913 is effect of pore pressure 1198914 is effectof differential pressure 1198915 is effect of drill-bit diameter andWOB1198916 is effect of rotary speed1198917 is effect of drill-bit toothwear and 1198918 is effect of bit hydraulic jet impact force

Equation (3) can be broken down into 8 subequations asfollows

1198911 = 1198902303lowast119886

1 (4)

1198912 = 1198902303lowast119886

2lowast(TVD

119873minusTVD) (5)

1198913 = 1198902303lowast119886

3lowastTVD069(EPPminusEPP

119873)

(6)

1198914 = 1198902303lowast119886

4lowastTVDlowast(EPPminusECD)

(7)

1198915 = [WOBsfODbit minus (WOBsfODbit)119905(WOBsfODbit)119873 minus (WOBsfODbit)119905

]

1198865

(8)

1198916 = [RPMsf(RPMsf)119873

]

1198866

(9)

1198917 = 119890minus1198867lowastℎ (10)

1198918 =[

[

119865119895

(119865119895)119873

]

]

1198868

(11)

For the given subequations normalization factors and coef-ficients are calculated being within lower and upper limits(boundary conditions) and their final values are yielded fromsimulations being related to specific scenarios where dataand information were gathered (see Tables 1 and 2)

21 Effect of Formation Strength (1198911) and (4) The expressionof 1198861 primarily represents the effect of formation strengthon the rate of penetration (ROP) it also represents someparameters such as drilled solids which have not beenseparately modeled yet

22 Effect of FormationCompaction onROP (1198912) and (5) Thisterm models the effect of compaction on ROP assuming anexponential decrease in ROP with depth in a normally com-pacted formation The effect of compaction on penetrationrate had always been set to influence de ROP referenced to anormally compacted formation at 10000 [ft]

23 Effect of Pore Pressure ROP (1198913) and (6) This termalso models the effect of compaction on ROP assumingexponential increase of ROPwith the increased pore pressuregradient High ROP is common in formations like sandstoneswhile low ROP is common in shale and limestonesThe lowerROP is mainly related to overburden stresses consequentlyadumbrating a more compacted and less porous interval

24 Effect of Differential Pressure (1198914) and (7) This termrepresents the effect of pressure differential across the bottomof the hole on ROP Increased bottom hole pressure can havea negative effect on ROP because cuttings can be held onbottom thus increasing the friction and teeth wear of thedrill-bits and also decreasing the hole cleaning efficiencyThe differential pressure is basically a consequence of any

Mathematical Problems in Engineering 3

Table 1 Normalization values [4 7]

Value Unit(TVD)119873 10000 [feet](EPP)119873 900 [poundsgallon](WOBsfODbit)119873 400 [kilopoundsinch](119865119895)119873 1000 [pounds](RPMsf)119873 100 [rotation per minute]

Table 2 Drillability coefficients [4 7]

Lower boundary [unitless] Upper boundary [unitless]1198861 05 191198862 0000001 000051198863 0000001 000091198864 0000001 000011198865 05 201198866 04 101198867 03 151198868 03 06

increased mud densities with the given depth which affectsthe equivalent mud density and further the hydraulic col-umn pressure just on the bottom of the hole yielding toa differential overbalance pressure increase acting againstthe hole cleaning The effect of differential pressure hadbeen referenced to 90 [ppg] as the limit in pressure to beconsidered as equalized

25 Effect of Drill-Bit Diameter and WOB (1198915) and (8) Thisterm models the effect of drill-bit weight and diameter onpenetration rate Increased WOB has an exponential andproportional response on ROP It should be noted that thedrilling process starts after a minimal applied load (WOBthreshold) Normally after a maximum WOB (founderingpoint) it starts influencing negatively the ROP basically dueto the less hydraulics fed into the system which might notbe high enough to clean the cuttings in the same efficiencyas new hole is developed Figure 1 gives an overview ofthis behaviour in which the foundering can be delayed andshifted to a higher value of applied WOB by reengineering(ie improving hydraulics) which is represented by (11) Itis also important to highlight that the penetration rate isdependent on the hole size where the ROP has an inverselyproportional relation to the drill-bit sizes (presalt sectionsrange normally from 8510158401015840 to 122510158401015840) The relation based onpast field experience had always been normalized to be equalto 10 for 4000 [lbfin] of WOB over drill-bit diameter

26 Effect of Rotary Speed (1198916) and (9) This term representsthe effect of rotary speed on ROP It assumes that theincreased rotary speed is directly proportional and expo-nential to the penetration rate Normally after a maximumincreased RPM (foundering point) it has a negative effect onROP vibration can also be a responsible factorThis term hadbeen normalized to be equal to 10 for 100 [rpm]

Inflection point

WOB

ROP

Performance is enhanced byredesigning to extend the

foundering point

Potential performance enhancementRegion III founder

Region II efficient drill-bit

Region I inadequate depth-of-cut (DOC)WOB threshold

Figure 1 Traditional drill-rate curve highlighting optimum andmaximumWOB regions (region II) [13]

27 Effect of Drill-Bit Tooth Wear (1198917) and (10) The drill-bitsare basically tools designed to produce a generally cylindricalhole and are broadly classified into two main types accordingto their primary cuttingmechanism the roller-cutter-bit typeand the fixed-cutter-bit or drag-bit type The first one appliesa mechanism of fracturing or crushing the formation andconsists of two or more cones (normally three) which havethe cuttings elements attached to it and rotate about the axisof the cones as the drill-bit rotates at the bottom of the holeThe second type has a mechanism of scraping (shearing)or grinding the formation being drilled having as cuttingelements normally natural or synthetic diamond (PDCdrill-bits) and consisting of fixed cutters in the blades that areintegratedwith the body of the drill-bit which rotates as a unitwith the drill-string

This term represents the effect of the tooth wear ofthe drill-bit on penetration rate The value of 1198867 dependsprimarily on the drill-bit type and less on the geology of theformation When carbide insert bits are used penetrationrate does not vary significantly with tooth wear Milledtooth bits have more severe penetration rate decrease withwearing Bourgoyne and Young defined the partial toothwear equation as given in (8) Tooth wear is affected byformation abrasiveness tooth geometry bit weight rotaryspeed and the cleaning processes It should be noted that asit is a general term in carbonate formations like presalt lay-ers polycrystalline-diamond-compact (PDC) bits are mostlyused

28 Effect of Bit Hydraulic Jet Impact Force (1198918) and (11)There are several types of drilling fluids widely used inthe petroleum industry each one having its specific designalong the actual implicit needs They are mainly divided intofour main system types freshwater saltwater oil-synthetic-based and pneumatic The main factor affecting either oneappliance is related to the final applicability purpose costsenvironmental impact and required technical performanceThe most frequently applied type for presalt operations is thesynthetic-oil-based mud (SOBM) Independently from thechosen design they have major impact on the penetrationrate either by the solids content or by the overbalance applied


(7) and also by the jet impact force acting just crossing thedrill-bit or more specifically the drill-bit nozzles

The last term models the effect of bit hydraulics onpenetration rate Increased jet force implies better cleaningof cuttings around the drill-bit teeth on the bottom of thehole and also better hydraulic environment for cutting trans-portation to the surface bymaintaining thewhole area aroundthe drill-bit and drill-string more clean avoiding differentialsticking and decreasing the friction rateWarren (1981) foundin his microbit experiments that ROP is proportional to aReynolds number group and that increasedReynolds numbercan increase the ROP [6] Moreover McLean (1964) andWarren (1984) experimentally showed how the drill-bit jetimpact force could positively influence the ROP This termhad been normalized to be equal to 10 for a jet impact of1000 [lbf] [16ndash18]

3 Practical Application of the Model

Since 1974 when the model was originally publishedmeasure-while-drilling (MWD) and logging-while-drilling(LWD) tools became mainstream scientists and engineershave better understanding of downhole conditions and costreduction is more evident than before driving to the factthat a better understanding of BYM could help the industrymoving forward Better understanding of the model could beachieved by a practical application and investigation

For a better understanding of the BYM model andidentification of room for improvements andor alterationof subfunctions already published academic sources andsimulations were investigated and several simulations wereran in presalt layers where actual field data were available Itis important to highlight that even the model was developedwith the aim of optimizing ROP in more soft formations(compared to carbonate) and also with the utilization ofroll-cutter-bits there is still a way to show its applicabilityin different formations by changing the main normalizationfactors and coefficients boundaries since these properties arecrucial andwith adjustments on the boundary conditions canhelp in the development process

31 Presalt Layers Thedecreasing production rate and grow-ing demand for hydrocarbons lead the petroleum industry toexplore reserves inmore challenging environmentOne of theplaces where hydrocarbon supply can be secured on the longrun is located close to the shores of Brazil and on the otherside of the Atlantic Ocean close to the shores of AngolaThisenvironment is known as presalt formations (Figure 2) [14]

The presalt denomination is used to designate carbon-ate geologic layers that were formed before the evaporitesdeposition which has been accumulated above the carbonatelayers themselves Earlier the hydrocarbon reserves whichwere found above the evaporites layers started to run outand consequently prospections of oil and gas in presalt layersgained interest In 2006 in Brazil a great reserve of oil andgas was found in a basin in similar conditions of layers thatextends for approximately 800 [km] offshore between thecity of Vitoria (State of Espirito Santo) and Florianopolis(southern region of Brazil) It is likely that the greatest presalt

reserves in the world are found there in the northeast tosouth of Brazil in the Gulf of Mexico and in Africarsquos WestCoast close to the shores of Angola [14]These presalt clustersstructure (from Brazil and Africa) were developed around160 million years ago from the separation of the continentalsuperstructure Gondwana (part of Pangea supercontinent)into the American and African continents Figure 3 shows asimilarity indication inwhich the presalt carbonate geologicalformations from Brazilian Santos and Campos Basin arelocated relatively close to the shores of South America beingalso evidenced on the African side since South America andAfrica have common geological history

The rifting created the conditions which were necessaryfor the deposition of sediments sea water (secured lowenergy and high salinity environment) started to fill upthe space between the formerly attached continents whichfavorably influenced the development of bacteria coloniesThe secretion of these bacteria allied with the precipitationof carbonate salts created nuclei for the formation of micro-bialites (carbonate rocks) on which hydrocarbons couldaccumulate Normally most accumulations in these regionshave presalt origin but in some cases the salt slips and opensway for them to migrate allowing them to accumulate inthe postsalt rocks Even though they have the same originthey may have some differences that is in the postsalt casebacteria could consume the lighter part of the oil fromwhich gasoline and diesel are extracted For the presalt oil ahigh reservoir of rocks such as coquinas and volcaniclasticsallied with a higher temperature above 80 [∘C] (given thegreater depths) creates a condition that sterilizes the oil andpreserves its qualities thus lighter oil and gas are expected tobe found [14] This phenomenon can be especially importantfor Brazil since its already discovered reserves were mainlyheavy oil reserves which are not beneficial for refineriesto prepare diesel fuel and therefore in the last couple ofdecades crude and other refined petroleum products wereimported to Brazil Nevertheless light oil or natural gasexploration and extraction from presalt layers in the Santosand Campos basins could have an impact for changing thissituation

As it was emphasized in the introduction exploration ofpresalt layers is a highly challenging enterprise The maindifficulties are operations in ultradeep water (1500 [m] anddeeper) reservoir locations deeper than 5000 [m] expansionover large areas with high gas-oil ratio high pressure andlow temperature and lying below thick evaporites layers(approximately 2000 [m] thick in some regions) mainlylocated offshoremdashup to 300 [km] off the coast with harshoceanic conditions Table 3 highlights some characteristics ofthese carbonate rocks from different publications

32 Simulation with Originally Suggested Drillability Coef-ficients and Normalization Factors Simulation were estab-lished based onpresalt actual drilling field dataTheoriginallysuggested starting coefficients and normalization values wereused with the help ofMicrosoft Excel 2010 andOracle CrystalBall Version 1122 softwares Crystal Ball is a MicrosoftExcel add-on application for predictivemodeling forecastingsimulation and formulation optimization The input data


Table 3 Main rock properties of presalt carbonates [8ndash11]

Author Publication year Porosity [] Permeability [mD] Approximate depth [m] Field nameLower value Upper value Lower value Upper value

Carminatti et al [8] 2008 9 12 100 5000ndash6500 Sao Paulo PlateauMello et al [9] 2011 2 15 NA NA 3000ndash5500 BuracicaPicarrasMello et al [9] 2011 2 14 NA NA 3000ndash5500 JiquiaTapemaJohann et al [10] 2012 9 12 100 5000ndash6500 NA

Remotely located offshoreexploration area up to

300km offshore

Total depth from5000 to 7000

metersW

ater

dep

th2000

m3000

m

Salt layer more than2000 meters thick

Presalt hydrocarbonreservoirs

Presalt layer newexploratory frontier

Posts

alt f

ocus

until

2006

Figure 2 Schematic of a presalt layer location and its surroundings [14]

used were the originally suggested drillability coefficients(with lower and upper boundaries) and the actual measuredROP values Crystal Ball recomputed the drillability coeffi-cients for the best possible fitting to actual field ROP data(targeting 119877-squared to 1) while normalization factors wereunchanged Table 4 and Figure 4 detail the first results Thegraph represents that the results are only depth-based (nottime-based)119877-squared enables researchers to test hypotheses or pre-

dict future outcomes by statistically measuring how close theoriginal data are to the fitted regression line In regressionthis parameter is a statistical measure of how well theregression line approximates the real data points

119877

2= 1 minus

sum119894 (ROPfield 119894 minus ROPcalc 119894)2

sum119894 (ROPfield 119894 minus ROPfield)2 (12)

Relative error has also been calculated for results comparisonpurposes Relative error is the ratio of an error in a measuredor calculated quantity to the magnitude of that quantity

Relative Error =sum1198941003816100381610038161003816

(ROPfield 119894 minus ROPcalc 119894)1003816100381610038161003816

ROPfield (13)

The results show adequate values however from the 119877-squared meaningful conclusions cannot be derived sincethey are out of the normally accepted range [0 to 1] (minus194)Nevertheless publications have shown that 119877-squared is notthe most reliable solution to evaluate results of nonlinearregression trends However in the same time relative errorshowsmore adequate results with the value of 050 (Figure 4)Considering the fact that the119877-squaredwas not a reliable wayto measure the fitting another simulation was run with theaimof getting the relative error as close as possible to 0 insteadof 119877-squared as close as possible to 1 (Table 5 and Figure 5)


WaterPostsalt marine shalesImpermeable salt layer

up to 2km thick

Oil fields

Porouslimestones

and dolomitesPorous lakesandstones

Highly organic oilgenerating lake

shales


shalesOil

AngolaSpreadingcenterBrazil

Figure 3 Presalt occurrences schematic emphasising Brazil andAngola [15]

ROP analysis ROP field versus Bourgoyne and Young coefficients ROP model calculated

ROP5 field dataROP calculated

241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811

Drilling mechanics samples (unitless)

0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

Figure 4 Result graph of the simulation with originally suggesteddrillability coefficients and normalization values with the depth-based ROP plotting

ROP analysis ROP field versus Bourgoyne and Youngcoefficients ROP model calculated


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


Figure 5 Result graph of the simulation with originally suggesteddrillability coefficients and normalization values with the aim ofminimizing relative error with depth-based ROP plotting

Table 4 Results of the simulation with originally suggested drilla-bility coefficients and normalization values

Lowerboundary[unitless]

Upperboundary[unitless]

Calculatedcoefficients[unitless]

1198861 05 19 09868661198862 0000001 00005 00000011198863 0000001 00009 00009001198864 0000001 00001 00000011198865 05 20 05000001198866 04 10 07029631198867 03 15 03000001198868 03 06 0599998119877-squared minus19388546Relative error 05067288

Table 5 Results of the simulationwith originally suggested drillabil-ity coefficients and normalization values with the aim ofminimizingthe relative error





Based on the obtained values the relative error showsmore adequate matching than in case of the first simulationwith 046 on top of 051 119877-squared shows minus261 comparedto minus194 confirming the unreliability of 119877-squared basedevaluation Based on the experience and gained knowledgeof the two simulation runs a third one was established toinvestigate the behaviour of the BYM with different startingparameters

33 Simulation with Modified Drillability Coefficients andNormalization Factors From the originally published nor-malization factors and parameters other academic sourceswere investigated to verify how the original model has beenalternated over the years It is important because understand-ing the development way of the BYM can help to establish abetter fitting simulation and identify possible ways for furthermodel improvement allowing room for consistent changesuggestions Table 6 summarizes the most important param-eters gathered (normalization values drillability coefficientsand 119877-squared values) final results can be observed later inFigure 6


Table 6 BYM applied parameters from different academic sources

Parameter Normalizationfactor

Lowerboundary

Upperboundary

Bourgoyne Jr and Young Jr 1974 and 1986 [4 7](RPMsf)119873 100 [rpm] mdash mdashTVD119873 10000 [ft] mdash mdashEPP119873 900 [ppg] mdash mdash(WOBsfODbit)119873 400 [k-lbfin] mdash mdash(119865119895)119873 1000 [lbf] mdash mdash1198861 mdash 05 191198862 mdash 0000001 000051198863 mdash 0000001 000091198864 mdash 0000001 000011198865 mdash 05 201198866 mdash 04 101198867 mdash 03 151198868 mdash 03 06119877-squared value mdash NA

Eren - 2015 [11](RPMsf)119873 60 [rpm] mdash mdashTVD119873 8000 [ft] mdash mdashEPP119873 900 [ppg] mdash mdash(WOBsfODbit)119873 400 [k-lbfin] mdash mdash(119865119895)119873 1000 [lbf] mdash mdash1198861 10005588 3291428571198862 0000191 00047911198863 000035 06588511198864 0000057 00003471198865 0102882 08528571198866 048 16842921198867 0284286 25873061198868 minus063243 1080511119877-squared value 0379 05395

Irwan et al - 2012 [12](RPMsf)119873 100 [rpm] mdash mdashTVD119873 10000 [ft] mdash mdashEPP119873 900 [ppg] mdash mdash(WOBsfODbit)119873 400 [k-lbfin] mdash mdash(119865119895)119873 1000 [lbf] mdash mdash1198861 3911198862 000009451198863 000006861198864 00008641198865 0371198866 2231198867 00251198868 067119877-squared value NA



241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

Figure 6 Result depth-based ROP graph of the simulation withauthor-suggested values

Based on the already obtained results from the first andsecond simulations and the results of other authors a newset of starting coefficients and normalization factors areproposed with the aim of improving fitting the field and thecalculated ROP more adequately (Tables 7 and 8)

It can be observed that the relative error value (0266) andalso the graphical response (Figure 6) of the simulation (sothe calculated ROP) values are fitting the actual measuredfield data much more consistently than in case of the firsttwo simulationsThese more reliable results are related to thechanged coefficient values and normalization factors Despitethe better fitting there is still room for improvements andvalidation of the results on more field data

4 Summary and Conclusion

Drilling operations in presalt layers is a challenge for thepetroleum industry in many perspectives One of the mostimportant perspectives from both technical and economicsides is the decreased ROP Increasing the ROP can beone of the keys to success so deeper understanding of theongoing processes properties and driving mechanisms ofthe ROP is crucial The Bourgoyne and Young ROP modelis a viable approach to understand and calculate the above-mentioned details but further considerations data miningand recurrent investigation are necessary

The simulations in this paper show improving resultsconsidering presalt drilling data but more simulations andactual field data are necessary for further development Alter-ation of the original model that is adding more functionsfor example vibrations changing range of coefficients rangeand normalization factors can also be a way for furtherimprovements as it is stated in the publication It is importantto highlight that the BYM was not specifically developed forPDC but roller-cutter-bit type it has adequately respondedand showed reliable simulation results Nevertheless furtheradjustment of the model could lead to better match ofsimulation and real life results

It was conclusive that relative error enabled the evaluationof the simulation and real life datamore adequately thanwhat


Table 7 Current authorsrsquo proposed values 2015


Lowerboundary

Upperboundary

(RPMsf)119873 60 [rpm] mdash mdashTVD119873 11200 [ft] mdash mdashEPP119873 900 [ppg] mdash mdash(WOBsfODbit)119873 400 [k-lbfin] mdash mdash(119865119895)119873 1000 [lbf] mdash mdash1198861 mdash 05 3911198862 mdash 0000001 00047911198863 mdash 0000001 06588511198864 mdash 0000001 00008641198865 mdash 0102882 201198866 mdash 04 2231198867 mdash 0025 25873061198868 mdash 03 1080511

Table 8 Results of the simulation with author proposal aiming tominimize the relative error



CalculatedCoefficients[unitless]

1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461

119877-squared allowed Moreover by applying a wider range ofapplicable drillability coefficients and normalization factorsthe relative error showed improvements from 46 down to27 a gain of about 20

Nomenclature

TVD True vertical depth [feet]TVD119873 True vertical depth normalization

value [feet]EPP Actual equivalent pore pressure

gradient [poundsgallon]EPP119873 Actual equivalent pore pressure

normalization value[poundsgallon]

ECD Actual equivalent circulatingdensity [poundsgallon]

WOBsf Surface measured weight on bit[kilopounds]

ODbit Drill-bit outside diameter [inches](WOBsfODbit)119873 Weight on bit over drill-bit out-

side diameter normalization value[kilopoundsinch]

RPMsf Drill-string surface measured rota-tional speed [rotation per minute]

RPMsf119873

Drill-string surface measured rota-tional speed normalization value[rotation per minute]

ℎ Drill-bit grading fractional toothwear [unitless]

119865119895 Hydraulic jet impact force beingapplied beneath the drill-bit[pounds]

1198861 Formation strength and drillingfluid properties coefficient[unitless]

1198862 Normal compaction trend coeffi-cient [unitless]

1198863 Undercompaction and pore pres-sure coefficient [unitless]

1198864 Differential pressure coefficient[unitless]

1198865 Constant dependent on drillingconditions and WOB and drill-bitcurve behavior [unitless]

1198866 Constant dependent on drillingconditions and RPM curve behav-ior [unitless]

1198867 Tooth wear coefficient [unitless]1198868 Hydraulic coefficient [unitless]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to disclose gratitude to the Chairof Drilling and Completion Engineering from the Mon-tanuniversitat Leoben Austria to the Coordenacao deAperfeicoamento de Pessoal de Nıvel Superior (CAPES) andto the Agencia Nacional do Petroleo Gas Natural e Biocom-bustıveis (ANP) by means of the BEX 050615-0 and PRH48-ANPMCTI

References

[1] R A Cunningham Laboratory Studies of the Effect of RotarySpeed on Rock-bit Performance and Drilling Cost AmericanPetroleum Institute 1960

[2] W C Maurer ldquoThe lsquoPerfectmdashcleaningrsquo theory of rotarydrillingrdquo Journal of Petroleum Technology vol 14 no 11 1962

[3] E M Galle and H B Woods Best Constant Weight and RotarySpeed for rotary Rock Bits American Petroleum Institute 1964

[4] A T Bourgoyne Jr and F S Young Jr ldquoA multiple regressionapproach to optimal drilling and abnormal pressure detectionrdquo


Society of PetroleumEngineers Journal vol 14 no 4 pp 371ndash3841974

[5] M G Bingham A New Approach to Interpreting Rock Drillabil-ity Petroleum Publishing Company 1965

[6] T M Warren ldquoDrilling model for soft-formation bitsrdquo Journalof Petroleum Technology vol 33 no 6 1981

[7] A T Bourgoyne JrApplied Drilling Engineering Handbook vol2 Society of PetroleumEngineers (SPE) Richardson Tex USA1986

[8] M Carminatti B Wolff and L Gamboa ldquoNew exploratoryfrontiers in Brazilrdquo in Proceedings of the 19th World PetroleumCongress (WPC rsquo08) WPC-19-2802 Madrid Spain June-July2008

[9] M RMello A A Bender N C A Filho and E DeMio ldquoGiantsub-salt hydrocarbon province of the greater campos BasinBrazilrdquo in Proceedings of the Offshore Technology ConferenceOTC Brasil Rio de Janeiro Brazil October 2011

[10] P R Johann A F Martini A Maul and J P P Nunes ldquoReser-voir geophysics in Brazilian pre-salt oilfieldsrdquo in Proceedings ofthe Offshore Technology Conference (OTC rsquo12) OTC-23681-MSHouston Tex USA April 2012

[11] T Eren Real-time-optimization of drilling parameters duringdrilling operations [PhD thesis] Middle East Technical Univer-sity 2015

[12] S Irwan ldquoOptimization of weight on bit during drilling oper-ation based on rate of penetration modelrdquo Journal UniversitasPasir Pengaraian vol 4 no 1 2012

[13] F E Dupriest and W L Koederitz ldquoMaximizing drill rateswith real-time surveillance of mechanical specific energyrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Amsterdam The Netherlands February2005

[14] I Waisberg ldquoBrazilrsquos Pre-Salt Layerrdquo University of Stanford2015 httplargestanfordeducourses2011ph240waisberg1

[15] T David and A Knascimento Pre-salt occurrences close to theshores of SouthAmerica andAfrica 2015 httpwww2b1stcon-sultingcomwp-contentuploads201210Petrobras Pre-saltSantos-basin Campos-Basin Esperito-Santo-Basin Mapjpg

[16] J R Eckel ldquoMicrobit studies of the effect of fluid properties andhydraulics on drilling raterdquo Journal of PetroleumTechnology vol19 no 4 pp 541ndash546 1967

[17] R H McLean ldquoCrossflow and impact under jet bitsrdquo Journal ofPetroleum Technology vol 16 no 11 1964

[18] T M Warren and W J Winters ldquoThe effect of nozzle diameteron jet impact for a tricone bitrdquo Society of Petroleum EngineersJournal vol 24 no 1 pp 9ndash18 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Young Jr (1974) and Warren (1981) actual and subsequentmodels were mainly based on further improvements on topof just-mentioned root ones [1ndash6]

Apart of the BYM ROPmodel detailed in the subsequentchapters the next equations below by Galle and Woods

(1964) model (1) andWarren (1981) model (2) are shown as amodel comparison reference and not as the main focus of thepaper

ROP = 119862119891 sdotWOBsf

119896sdot [119890

(minus100RPMsf2)sdot RPMsf

1198871+ 1198872 sdot RPMsf (1 minus 119890

(minus100RPMsf2))]

(0928125 sdotOD2bit + 6 sdotODbit + 1)119901

(1)

ROP = 119870 sdotRPMsf sdot (WOBsf minus (WOBsf)119905)

2

OD2bit sdot 1198782 (2)

where 1198871 is 075 or 0428 for soft and hard formationsrespectively [unitless] 1198872 is 05 or 02 for soft and hardformations respectively [unitless] 119901 is tooth wear coefficient[unitless] 119896 is coefficient accounting for WOB influence onROP [unitless] 119862119891 is coefficient accounting for bit typehydraulics drilling fluid and formation strength [unitless]and 119870 is constant dependent on drill-bit dullness and for-mation abrasiveness as a consequence of drilling conditions[unitless]

Research and development continued yet one of themostcomprehensive ROP models is the one proposed by Bour-goyne Jr and Young Jr published in 1974 which considersthe effects of the depth the characteristics of the formationbeing drilled the drill-bit size the mechanical factors ofthe drilling process (ie WOB and RPM) and the mudsystem properties allowing each one to be adjusted by fittingcoefficients [4 7]

2 BYM ROP Model

The modelrsquos main equation (3) consists of 8 subfunctionswhich act and have significant influence on the ROP perfor-mance [2]

ROP = 1198911 lowast 1198912 lowast 1198913 lowast 1198914 lowast 1198915 lowast 1198916 lowast 1198917 lowast 1198918 (3)

where 1198911 is effect of formation strength 1198912 is effect of depthand compaction 1198913 is effect of pore pressure 1198914 is effectof differential pressure 1198915 is effect of drill-bit diameter andWOB1198916 is effect of rotary speed1198917 is effect of drill-bit toothwear and 1198918 is effect of bit hydraulic jet impact force

Equation (3) can be broken down into 8 subequations asfollows

1198911 = 1198902303lowast119886

1 (4)

1198912 = 1198902303lowast119886

2lowast(TVD

119873minusTVD) (5)

1198913 = 1198902303lowast119886

3lowastTVD069(EPPminusEPP

119873)

(6)

1198914 = 1198902303lowast119886

4lowastTVDlowast(EPPminusECD)

(7)

1198915 = [WOBsfODbit minus (WOBsfODbit)119905(WOBsfODbit)119873 minus (WOBsfODbit)119905

]

1198865

(8)

1198916 = [RPMsf(RPMsf)119873

]

1198866

(9)

1198917 = 119890minus1198867lowastℎ (10)

1198918 =[

[

119865119895

(119865119895)119873

]

]

1198868

(11)

For the given subequations normalization factors and coef-ficients are calculated being within lower and upper limits(boundary conditions) and their final values are yielded fromsimulations being related to specific scenarios where dataand information were gathered (see Tables 1 and 2)

21 Effect of Formation Strength (1198911) and (4) The expressionof 1198861 primarily represents the effect of formation strengthon the rate of penetration (ROP) it also represents someparameters such as drilled solids which have not beenseparately modeled yet

22 Effect of FormationCompaction onROP (1198912) and (5) Thisterm models the effect of compaction on ROP assuming anexponential decrease in ROP with depth in a normally com-pacted formation The effect of compaction on penetrationrate had always been set to influence de ROP referenced to anormally compacted formation at 10000 [ft]

23 Effect of Pore Pressure ROP (1198913) and (6) This termalso models the effect of compaction on ROP assumingexponential increase of ROPwith the increased pore pressuregradient High ROP is common in formations like sandstoneswhile low ROP is common in shale and limestonesThe lowerROP is mainly related to overburden stresses consequentlyadumbrating a more compacted and less porous interval

24 Effect of Differential Pressure (1198914) and (7) This termrepresents the effect of pressure differential across the bottomof the hole on ROP Increased bottom hole pressure can havea negative effect on ROP because cuttings can be held onbottom thus increasing the friction and teeth wear of thedrill-bits and also decreasing the hole cleaning efficiencyThe differential pressure is basically a consequence of any









Inflection point

WOB

ROP


foundering point

























300km offshore


metersW

ater

dep

th2000

m3000

m




Posts

alt f

ocus

until

2006




119877

2= 1 minus






ROPfield (13)




up to 2km thick

Oil fields

Porouslimestones



shales


shalesOil





241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)




0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


















Lowerboundary

Upperboundary






241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)











Lowerboundary

Upperboundary






1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461


Nomenclature










RPMsf119873













Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Inflection point

WOB

ROP


foundering point

























300km offshore


metersW

ater

dep

th2000

m3000

m




Posts

alt f

ocus

until

2006




119877

2= 1 minus






ROPfield (13)




up to 2km thick

Oil fields

Porouslimestones



shales


shalesOil





241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)




0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


















Lowerboundary

Upperboundary






241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)











Lowerboundary

Upperboundary






1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461


Nomenclature










RPMsf119873













Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























300km offshore


metersW

ater

dep

th2000

m3000

m




Posts

alt f

ocus

until

2006




119877

2= 1 minus






ROPfield (13)




up to 2km thick

Oil fields

Porouslimestones



shales


shalesOil





241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)




0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


















Lowerboundary

Upperboundary






241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)











Lowerboundary

Upperboundary






1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461


Nomenclature










RPMsf119873













Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











up to 2km thick

Oil fields

Porouslimestones



shales


shalesOil





241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)




0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)

241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


















Lowerboundary

Upperboundary






241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)











Lowerboundary

Upperboundary






1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461


Nomenclature










RPMsf119873













Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Lowerboundary

Upperboundary






241

371

361

351

341

331

191

251

171

291

271

261

131

141

151

161

231

391

311

321

381

301

401

211

201

181

121

281

111

101

221

41

716151

31

91

2111

811


0

2

4

6

8

10

12

14

16

18

20

ROP

(fth

r)











Lowerboundary

Upperboundary






1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461


Nomenclature










RPMsf119873













Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Lowerboundary

Upperboundary






1198861 05 391 08491651198862 0000001 0004791 00000011198863 0000001 0658851 00000011198864 0000001 0000864 00000011198865 0102882 20 01028821198866 04 223 04000001198867 0025 2587306 00250001198868 03 1080511 1080511Relative error 02661461


Nomenclature










RPMsf119873













Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article mathematical modeling applied to drilling...

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