Stochastic Adaptive Learning Rate in an Identification Method: An Approach for On-line Drilling Processes Monitoring

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1 9 Amerian Control Conerene Hyatt Regeny Riverront, St. Louis, MO, USA June -, 9 FrB3. Stohasti Adaptive Learning Rate in an Identiiation Method: An Approah or On-line Drilling Proesses Monitoring A. BA, S. Hbaieb, N. Mehbal, and M. Vergé Abstrat On-line drilling proesses monitoring is an essential task in enhaning their perormanes. In oilield industry, dysuntions that might our have to be deteted at the earliest possible stage in order to preserve drilling eiieny. his paper deals with a methodology or drilling proesses monitoring by identiying time varying parameters. he basi idea behind the proposed algorithm is to improve the traking ability o parameters hange by means o an identiiation method using a new approah to adjust the orgetting ator. he eetiveness o the developed method is highlighted through experimental data obtained rom tests ampaign. I. INRODUCION HE neessity to disover petroleum in hostile environments, justiied by the derease o world s oil reserve, attrats a great deal o interest in terms o sientii investigation and drilling proesses (Fig..) advanements, or eonomi suess. In the ontext where the daily drilling rig ost an exeed 5. USD, drilling proesses monitoring beome important. Its beneits ould improve operational perormanes as well as preditive maintenanes. Nevertheless, one o the major requirements or eiient real-time monitoring is to reeive and proess data quikly in order to make a useul deision. his requirement is possible with the new progress made in telemetry tehnology that enables data transer at requenies exeeding KHz. In addition, a better understanding o drilling proesses behavior ombined with the knowledge o a part o their theoretial modeling, identiiation and statistial signal proessing [], [] are opening new ways or drilling proesses monitoring [3]. However, drilling proesses remain ompliated systems and roks being drilled are more oten badly known. As a result, a model whih takes into aount aurately both proess and rok is diiult to ahieve and represents a researh topi o grows interesting. During drilling, luid irulation plays a ruial role. It redues the rok hardness by hydration proess and allows uttings transportation rom downhole to surae. When uttings proess proeeds, a bit balling dysuntion (Fig..) an our. his senario happens when uttings and mud (drilling luid) stuk on the teeth o the bit (head o the proess) and annot be washed by the luid irulation. At a ertain stage, the bit beomes ompletely overed; onsequently the utting proess annot be aomplished as expeted [4]. For this reason, several studies onerning bit balling redution were arried out and some results were ound. Among them: drill bit design optimization, suitable drilling parameters and the best drilling mud hoie are mostly used as solutions by drilling engineers. However, the main drawbak o these strategies results rom the lak o knowledge onerning the rok to drill beore drilling operations. o overome this diiulty, bit-rok interation model has been developed [5]. he speiiity o the ahieved model lies in the possibility o not involving rok properties. Based on this model, a real-time approah or bit balling is developed in this paper. he methodology developed here alls into the ategories o reursive least square algorithm (RLS) []. Note that, one o the noteworthy advantages o RLS arises out o the apability that it traks time varying parameters by disarding old data and giving more weight to reent ones through a orgetting ator. In this ase, the auray o the estimated parameters depends greatly on the hoie o the orgetting ator. It is wellknown that, the lowering the orgetting ator inrease the rate o alse alarm and the inreasing the orgetting ator inrease the lak o detetion. o avoid suh situations, a lot o eorts have been made to make the orgetting ator adjustable [6], [7]. A. BA. is with both Laboratoire de Méanique des Systèmes et des Proédés (UMR-CNRS), Eole Nationale Supérieure d Arts et Métiers, 5, Boulevard de l hôpital, Paris, Frane and Shlumberger Riboud Produt Centre,, rue Henri Bequerel, 94 Clamart, Cedex, Frane (phone: ; ax: ; aba@ slb.om). S. Hbaieb. is with Shlumberger Riboud Produt Centre,, rue Henri Bequerel, 94 Clamart, Cedex, Frane (shbaieb@slb.om). N. Mehbal. and M. Vergé are with Laboratoire de Méanique des Systèmes et des Proédés (UMR-CNRS), Eole Nationale Supérieure d Arts et Métiers, 5, Boulevard de l hôpital, Paris, Frane (Nazih.mehbal@paris.ensam.r), (Mihel.Verge@paris.ensam.r). Fig.. Overall sheme o a drilling proess /9/$5. 9 AACC 537

2 It has been developed in [8] an algorithm whih adjusts the orgetting ator by using stohasti gradient desent method. In this method, the learning rate plays an essential role. Nevertheless, it is widespread that the quality o the gradient desent method is improved i the learning rate is variable. In this ontext, in addition to develop a methodology or bit balling detetion, it is the intent o this paper to extend the algorithm proposed in [8] by adjusting the orgetting ator through stohasti gradient desent method and making the learning rate adaptive [9]. he beneit o the proposed algorithm allows the orgetting ator to move along the diretion whih does not neessarily oinide with the gradient desent diretion []. hereby, the minimization proess is aelerated and the ast onvergene is ahieved. Eventually, it is also shown by this method that the traking ability is enhaned. A brie introdution and summary o the bit-rok interation model is pointed out in setion II. he proposed identiiation method is presented in setion III. In setion IV some experimental data in ase o bit balling are presented as well as the aurate estimate ahieved by using the developed methodology. In setion V onluding remarks are drawn. Fig.. Drill bit on the let without balling on the right with balling II. BI-ROCK INERACION MODEL Researh ativities regarding bit-rok interation modeling have been investigated and signiiant ontributions have been reported [5]. hese models aount or the utting ation o a single utter, whih is a Polyrystalline Diamond Compat (PDC). Cutting ation o eah utter onsists o two independent proesses, the utting proess and the rition proess. he torque and the weight W are deined by: = +, W = W + W where: a d = ε, W = ξ ε a d µ a γ W = π v, d = Ω he subsripts and denote utting and rition respetively. For desription o variables see ABLEI. he utting omponents and W orrespond to the ores transmitted by the utting ae o eah utter and the rition omponents and W orrespond to others ontats generated rom bit-rok interation. he response o the bit is obtained by ombining the utting and ritional proess [5]: W = ( β ) ε + µ γ a d a d with: β = µ γ ζ he speii energy whih orresponds to the neessary energy to grind a given volume o rok is deined by [5]: () E = E + µ γ S () where: E = ( β ) ε hrough () and () the ollowing relations have been obtained: W S =, a d E = a d It is lear rom () that the slope o E as a untion o S is only dependent on the bit shape ator and the rition oeiient. hereore, i the bit balling ours these parameters will be adversely aeted. As a onsequene, the bit balling is onsidered as a hange in this slope. hereby, our objetive amount to exploit the model () to develop a strategy or bit balling diagnosis. o ahieve this aim, we propose an identiiation method based on a new adjustment o the orgetting ator. his adjustment is made aording to the gradient desent method in the ase where the learning rate is adaptive. ABLE I VARIABLES USED Symbol Quantity Unity a Bit radius m d Depth o ut m E Speii energy Mpa S Drilling strength Mpa orque on bit Nm orque on bit, ut Nm orque on bit, rition Nm W Weight on bit N W Weight on bit, ut N W Weight on bit, rition N ε Intrinsi speii energy Mpa v Rate o penetration (ROP) m/h Ω Bit angular veloity rad/s µ Frition oeiient ξ Coeiient haraterizing utting ore γ Bit shape ator 538

3 A. RLS Identiiation III. IDENIFICAION MEHOD he eature o the developed algorithm is to enhane the traking ability in transient stage, by automatially shortening the value o the orgetting ator. It is perormed aording to the gradient desent method with an adaptive learning rate. We obtain an algorithm alled stohasti adaptive learning rate gradient variable orgetting ator in reursive least square ontext (SALR-GVFF-RLS). he proposed algorithm is an extension o the gradient variable orgetting ator reursive least square (GVFF-RLS) developed in [9]. he SALR-GVFF-RLS enables the orgetting ator to ahieve a ast desent owing to an adaptive learning rate whih aelerates the onvergene o the algorithm, when a transient stage happens. Consequently, the traking ability is improved. he advantage o the proposed method allows being able to diagnose the bit balling when it is inipient. Let us take again the equation () and rewrite it in a state spae representation: θ( = θ( y( = ϕ ( θ( + υ( Where n θ R is the parameter vetor to be estimated: θ ( µ γ θ ( = = θ ( E m y R, is the output vetor, here it is the speii energy E, υ( is the observation noise vetor, ϕ ( stands or the regressor vetor, and is onstruted rom drilling strength S. he lassi reursive least square estimation with orgetting ator (FF-RLS) is given by []: K( k P( k + ) = ε( = + ϕ ( P( ϕ( + ) = y( k [ P( K( ϕ ( P( ] + ) ϕ ( θ( θ( = θ( + K( ε( P( ϕ ( (3) Our objetive is to improve the adaptation quality by using a variable orgetting ator ( on the evidene o the gradient desent method as given by: = ( η ( (4) + In (4), η denotes the learning rate. However, it is wellknown that the perormanes o the gradient desent method depend on the hoie o the learning rate. Speially, its value needs to be in a ertain range so that to perorm a good onvergene. Moreover, it is not an easy task to hoose the best learning rate value. In this ontext, we propose a new algorithm that makes the learning rate adaptive onsequently to improve the eetiveness o the identiiation method in terms o traking ability. At eah stage, the determined learning rate depends on the urrent and the previous gradient value as shown in [9]. η = η( + α (, ( (5) + In this relation n, represents the usual inner produt in IR, α is alled the meta-learning rate. he gradient ( is obtained by adopting the ollowing methodology [8]: irst, the ollowing ost untion is minimized: J( k + ) = E ε [ ( ] In (6), E[ ] represents an expetation operator. hen, J( ( = ε( = E ε( he relationship (7) is obtained by dierentiating (6) with respet to. by deining: θ( ψ ( = we have : = ϕ ( ψ ( (6) (7) ε( (8) = [ y( ϕ ( θ( ] using (7) and (8) we obtain: ( = E[ ψ ( ϕ( ε( ] In aordane with [8] and ater some alulations, K ( an be rewritten as: K ( = P( ϕ ( thereore, the parameter vetor in the equation (3) beomes: θ ( = θ( + P( ϕ( ε( by deining: S ( k P( + ) = and, ater some alulations, the ollowing relation is 539

4 obtained: ψ ( = ( I K( ϕ ( ) ψ ( We have also: S( k + S( ϕ( ε( + ) = ( I K( ϕ ( ) S( ( I + ϕ( K K( K ( ) P( ( he SALR-GVFF-RLS algorithm an be summarized as ollows: K( k P( k ε( = θ( = θ( + K( ε( ( = [ ( + η( ψ η( = η( + α S( k + ) = + ) = ( + ϕ ( P( ϕ( y( k + ) = ( ( I K( ϕ ( ) S( ( I ( P( ϕ( [ P( K( ϕ ( P( ] + ) ϕ ( θ( (, ( ϕ( K ( ) ( P( + ( K( K ( ψ ( = ( I K( ϕ ( ) ψ ( + S( ϕ( ε( ( ϕ( ε( ] + In this algorithm, ( needs to be in the range o < (. he braket ollowed by the loor and the eiling + in the orgetting ator relation desribes the trunation that restrits the orgetting ator to the interval, ]. [ + IV. EXPERIMENAL DAA It is reognized in oilield industry that the main ause o bit balling dysuntion is due to inappropriate drilling parameters (ABLEII) ompared with rok properties and drill bit design. It has been intentionally hosen drilling parameters that ause bit balling dysuntion, in order to understand its eet on real data. hese measurements stand or the irst step or interpretation o bit balling behavior. ABLE II PARAMEERS DURING DRILLING ES Symbol Quantity Value and unity Q Flow rate 5 gpm v Rate o penetration 5 m/h Ω Bit angular veloity 3 rpm UCS Rok property 35 Mpa o understand how bit balling dysuntion an interere with the proess operation, measurements o E, S and rate o penetration (ROP) over time are given in Fig. 3. Beore sample number 5, ROP was adjusted ontinuously to ind a set o points that allows onstruting a slope. It an be notied that during this time interval S is almost onstant. It is due to the at that dysuntion has not yet ourred. In this ase, the mean value o the strength omes rom rok hardness. In spite o rok homogeneity and a onstant ROP in the sample interval between [45 5] S and E inrease. heir inrease omes rom and W and is due to the hange in the rition oeiient µ or/and the bit shape ator γ. hese hanges reveal a dysuntion on the bit and orrespond to the bit balling. However, in real drilling onditions it is not a straightorward task to diagnose this kind o aults by looking at the progress in measurements. It is mainly aused by the similar behavior viewed on S and E either in ase o hangeover rom slight rok to hard rok or the bit balling dysuntion. For this reason, we have developed an on-line identiiation method to determine the slope whih haraterizes E as a untion o S and onsequently deteting bit balling. ROP (m/h) E(Mpa) S(Mpa) 4 3 emporal measurement Fig. 3. Behavior o measurements in ase o bit balling A. Methodology adopted An early treatment o data was aomplished beore to start identiiation method. his treatment inludes elimination o missing values. his operation is made by ixing a threshold and omparing it to the data reeived at eah time step. Ater this, the data are iltered by a low pass ilter to redue the noise level. hen, the identiiation method is applied to data to haraterize the slope o E as a untion o S. At last, in order to generate automatially a ault indiator statisti Wald s tests were applied to the slope. he basi Wald s tests [] were hosen owing to its eetiveness. Moreover, Wald s tests are omputationally heap, very easy to implement and robust to modeling errors. he previous steps are illustrated in Fig. 4. o start identiiation method many initializations are needed. 54

5 v E, S E, S µ γ FF-RLS GVFF-RLS SALR-GVFF-RLS. Fig. 4. Monitoring method or bit balling detetion his inlude initial estimate o states, initial ovariane matrix, initial orgetting ator, initial learning rate and initial meta-learning rate. In this appliation, initialization o the orgetting ator is hosen aording to the bit balling progress. Correlating the initial value o the orgetting ator with the progress o bit balling behavior is useul in order to enable bit balling detetion in the sample interval rom whih its reversibility is oneivable by taking appropriate orretive ation. Ater several experimentations, it is interesting to point out, that the tuning parameters or the identiiation method developed here, depend diretly on drilling parameters. he best hoie o parameters at the initialization is the ollowing: ABLE III PARAMEERS A HE INIIALIZAION Symbol Value Signiiation θ ( ) Parameter vetor ψ ( ) Derivative parameter vetor with respet to P I Covariane matrix (). 97 Forgetting ator η () 5 Learning rate α 9 Meta learning rate B. Identiied slopes o emphasize the perormanes o the proposed algorithm a omparison o its traking apability with both the GVFF- RLS and the FF-RLS is ahieved. Slopes obtained rom the FF-RLS, the GVFF-RLS and the SALR-GVFF-RLS algorithms are shown in Fig. 5. In this igure and at the initialization, the parameter vetor, the ovariane matrix and the orgetting ator have the same values in all algorithms. Note that, in this igure, superior traking perormanes are obtained by the SALR-GVFF-RLS algorithm. However, at the starting up to samples, due to insuiient data enabling slopes to have a regular behavior, a high level o lutuations is present. In other words, this samples interval is required beore to obtain a good slope behavior. Here, whatever the developed method these lutuations are presents. hen, a transient regime happens and is the irst step toward a good identiiation method. At the sample 45 the slope starts to derease, this derease orresponds to the bit balling ourrene. It is shown in this igure that the proposed algorithm yields good Slope Fig. 5. Slopes progresses results during transient stage owing to its ast onvergene. For instane, during the samples interval between [45 35] the hange rom the saety operation to the bit balling dysuntion is ahieved more quikly by the SALR-GVFF- RLS. Note that booth algorithms onverge toward a same value. In the Fig. 6 the orgetting ator obtained by the GVFF-RLS and the SALR-GVFF-RLS are plotted. From this igure, the SALR-GVFF-RLS exhibits a aster desent. In addition, it is useul to point out that during the transient stage the adjustment o the orgetting ator is made automatially. However, at a ertain stage, it ould lead to several alse alarms beause o its low value. hen, it has been neessary to detet the end o the hange in the orgetting ator desent and to restore it to its initial value. his detetion is made by ixing a threshold and omparing it to the slope generated during the orgetting ator desent. One o the possible onsequenes o the orgetting ator reestablishment is the derease o lutuations generated in the slope beause o the low value o the orgetting ator outside the transient domain. During this study several data base have been tested. In all ases, the better traking is ahieved by the SALR-GVFF-RLS. GVFF-RLS SALR-GVFF-RLS Fig. 6. Forgetting ator progress 54

6 With the proposed method, the most sensitive tuning parameters are both learning rate and meta-learning rate. o perorm a good traking ability by applying our algorithm to experimental data, their values at the initialization need to be in the range o α 8 and 6 4 η (). he same values are used in all ases. In this study the sampling requeny was equal to 5 Hz. C. Deision to be taken o improve the reliability on deision to be taken a lassi Wald s tests [] were applied to data. Its parameters are determined while the system was in saety operation. It is started one the data are suiient and the slope having a regular behavior. In this ase, it proeeds rom sample number. In the Fig. 7. we present results obtained rom Wald s tests. Wald tests Fig. 7. Wald s tests results Wald tests FF-RLS Wald tests GVFF-RLS Wald tests SALR-GVFF-RLS It have been ahieved the irst, the seond and the last detetion respetively rom slopes obtained by the SALR- GVFF-RLS, the GVFF-RLS and the FF-RLS. he irst detetion is ahieved at the sample number 5. hen, the seond detetion at the sample number 555 and the last detetion at the sample number 6. he ABLEIV shows the results obtained by testing our data base and the number o samples neessary rom whih the detetion is made by Wald s tests. Here, there is no alse alarm, beause the Wald s test is airly robust to modeling errors and unertainties. In this study, the late in detetion is around 5 samples. It orresponds to the neessary time beore to take a deision. hrough the results reported here and other tests investigated, we have shown that with our methodology there is a gain higher than 5 samples, whih is a no negligible result in oilield area. ABLE IV WALD S ESS RESUL SALR-GVFF-RLS GVFF-RLS FF-RLS Data base Data base Data base Data base V. CONCLUSION In oilield industry, drilling proesses monitoring is an important issue. In this paper, an identiiation method with its apabilities to trak parameters hange by using a new strategy to adjust the orgetting ator aording to the SALR-GVFF-RLS has been developed. his method shows good results regarding bit balling detetion. Compared with both the GVFF-RLS and the FF-RLS it exhibits better perormanes in terms o traking ability during transient stage. he demonstration o the easibility o on-line bit balling detetion has been arried out by using several real data obtained rom tests ampaign. With the developed method a gain o at least 5 samples is ahieved. Moreover, this methodology remains valuable or diagnosis o others kinds o dysuntions that might our on the bit. his study stands or the irst step in the urther development and improvement o drilling proesses optimization. he requirement to diagnose bit balling when it is inipient is useul onerning an appropriate orretive ation to take and that brings the system to an optimal drilling ondition. For instane, it would enable to avoid lost o times, by leaning the drill bit more quikly. he next step o this study is to develop a ault tolerant ontrol algorithm in order to automatially adjust the input o the drilling proess in ase o bit balling detetion. ACKNOWLEDGMEN he authors would like to thank the Shlumberger Riboud Produt entre or supporting this work. Speial thanks are extended to the two anonymous reviewers or their omments helping us to improve the quality o the paper. REFERENCES [] L. Ljung, System Identiiation: heory or the User (nd). Prentie- Hall PR:Upper Sadle River, NJ [] M. Basseville and I. V, Nikiorov, Detetion o abrupt hanges: heory and appliation. Prentie-Hall, Englewood Clis, NJ, 993. [3] G. Nygaard and G. Naevdal, Nonlinear model preditive ontrol sheme or stabilizing annulus pressure during oil well, Journal o Proess Control, vol. 6, pp , Jannuary 6. [4] C. Cheatham and J. Nahm, Bit balling in water-reative shale during ull-sang rate test, Soiety o Petrolium and Engineers, exas, February, 99. [5]. Rihard and E. Detournay, Inluene o bit rok interation on stik slip vibrations o PDC bits, Soiety o Petroleum and Engineers, exas, Otober,. [6] L. Ljung and S. Gunnarsson, Adaptive traking in system identiiation-a survey, Automatia, 9, pp. 5-6, 99. [7] F. Gustason, Adaptive iltering and Change Detetion. Willey: WestSussex, England,. [8] M. B. Malik, State spae RLS with adaptive memory, Signal Proessing, vol. 86, pp , Marh 6. [9] L. B. Almeida,. Langlois, J. D. Amaral, and A. Plankhov, Parameter adaptation in Stohasti Optimization, In On-line Learning in Neural Networks, Cambridge University Press, pp. - 34, 998. [] J. Noedal and S.J. Wright, Numerial optimization: Springer