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elitist non-dominated sorting genetic algorithm. A Louisiana offshore field with abnormal formation pressure is considered for optimization. Several multi-objective optimization problems involving two- DQGWKUHHREMHFWLYHIXQFWLRQVZHUHIRUPXODWHGDQGVROYHGWR¿[RSWLPDOGULOOLQJYDULDEOHV7KHLPSRUWDQW objectives are: (i) maximizing drilling depth, (ii) minimizing drilling time and (iii) minimizing drilling cost with fractional drill bit tooth wear as a constraint. Important time dependent decision variables are: (i) equivalent circulation mud density, (ii) drill bit rotation, (iii) weight on bit and (iv) Reynolds number function of circulating mud through drill bit nozzles. A set of non-dominated optimal Pareto frontier is obtained for the two-objective optimization problem whereas a non-dominated optimal Pareto surface is obtained for the three-objective optimization problem. Depending on the trade-offs involved, decision makers may select any point from the optimal Pareto frontier or optimal Pareto surface and hence corresponding values of the decision variables that may be selected for optimal drilling operation. For minimizing drilling time and drilling cost, the optimum values of the decision variables are needed to be kept at the higher values whereas the optimum values of decision variables are at the lower values for the maximization of drilling depth. Drilling performance, rate of penetration, abnormal pore pressure, genetic algorithm, multi- Kew words: objective optimization drill bit rotating time, trip time, connection time, cost of 1 Introduction drill bits and rig cost. The drill bit rotating time depends on The technical cost of crude oil depends largely on the several drilling variables e.g., wellbore stability, class of expenditures due to exploration, development and production drill bit, weight on bit, rotary speed of drill bit, drilling mud operations. Among them, exploration expenditure is primarily properties, hydraulics of drilling mud, drill bit tooth wear and due to geological and geophysical surveys along with drilling activities. Development expenditure normally includes the the cost of drilling increases with depth in a parabolic manner cost of amortization of several wells that must be drilled to up to about 3,000 meters, and then exponentially increases produce hydrocarbon economically. The relative magnitude beyond 4,000 meters (Masseron, 1990). Therefore, marginal of exploration and development expenditure varies from improvement in drilling cost may reduce exploration and field to field and depends on the difficulties present in the GHYHORSPHQWFRVWVWRDVLJQL¿FDQWH[WHQW field. In general, exploration and development contribute To predict the rate of penetration (ROP) and the best set approximately 50%-80% of total expenditure. In this regard, of operating drilling variables, different drilling models are drilling also plays an important role and contributes 65%- proposed in terms of rock properties and drilling variables 80% of exploration and development cost. Expenditures are which are discussed in standard texts (Adams, 1985; Mitchell, also increasingly added to recover additional amounts of oil 1992; Bourgoyne et al, 2003). Graham and Muench (1959) and gas for given hydrocarbon in place. Details of exploration determined the optimum values of weight on bit (WOB) and and development expenditures for different oil wells have rotary speed (revolution per minute, rpm) of the drill bit to been discussed by Masseron (1990). The cost of drilling is minimize the drilling costs based on field data. The most usually expressed as the cost per footage or meterage drilled, common and popular ROP model used in drilling industry is and depends on type of rig used, geographic location and the d -exponent method (Bingham, 1964) which was based target drilling depth. Cost of drilling is also influenced by on rock mechanics principles. Jordan and Shirley (1966) modified Bingham’s equation and proposed drillability function which depends on depth and strength of formation. *Correspondence author. email: cguria.che@gmail.com Rehm and McClendon (1971) further modified Jordan and Received May 6, 2013 GULOOELWEHDULQJZHDU'HSHQGLQJXSRQWKHGULOOLQJGLI¿FXOWLHV 98 Pet.Sci.(2014)11:97-110 Shirley’s ROP equation by incorporating the actual weight reviewed by Deb (2001) and Coello Coello (2002). A popular of drilling mud. This ROP model has an ability to detect algorithm used for multi-objective optimization problems is and quantify abnormal pore pressure accurately. Bourgoyne the elitist non-dominated sorting genetic algorithm (NSGA- and Young (1974) developed a comprehensive nonlinear II) developed by Deb and his research group (Deb, 2001; Deb drilling model to predict drilling performance, abnormal et al, 2002). Nowadays, NSGA-II has been applied to solve formation pressure and optimal operating drilling variables. KLJKO\FRPSXWHULQWHQVLYHSUREOHPVLQWKH¿HOGRISHWUROHXP They performed multiple regression analysis to determine engineering related problems (gas lift optimization: Ray and the constants present in the drilling model using field data 5DVKLG6DUNHUZDWHUÀRRGLQJSHUIRUPDQFH+DQ and developed a relatively simple analytical procedure to et al, 2011; oil production planning: Singh et al, 2013). determine the best values of the operating drilling variables In this study, a model based on multi-objective drilling to minimize the cost of drilling. Reza and Alcocer (1986) optimization has been implemented using the elitist non- developed a dynamic nonlinear, multi-dimensional and dominated sorting genetic algorithm. The drilling model of dimensionless drilling model for deep drilling applications Bourgoyne and Young (1974) has been used to predict the ROP XVLQJWKH%XFNLQJKDPʌWKHRUHP%XFNLQJKDP and fractional tooth wear. Pore pressure variation with true They proposed three equations for the drilling model, YHUWLFDOGHSWK79' KDVEHHQSUHGLFWHGXVLQJWKH¿HOGGDWD namely, rate of penetration, rate of bit dulling and rate of of the Louisiana offshore formation to calculate the variation bearing wear involving several drilling variables for example of drilling depth and fractional wear loss of the drill bit with WOB, rotary speed of the drill bit, bit diameter, bit nozzle drilling time, and has been used for optimization studies. diameter, bit bearing diameter, drilling fluid characteristics Several multi-objective optimization problems are formulated SDUWLFXODUO\GHQVLW\DQGYLVFRVLW\ GULOOLQJÀXLGFLUFXODWLR Q involving conflicting objectives, namely, maximization of rate, differential pressure between formation and bottom WKH¿QDOGULOOLQJGHSWKPLQLPL]DWLRQRIWKH¿QDOGULOOLQJWLPH hole, rock hardness, bottom hole temperature and heat and minimization of the drilling cost. The drill bit tooth wear transfer coefficient. Maidla and Ohara (1991) developed an is the only constraint involved in the present optimization optimization technique based on Bourgoyne and Young (1974) studies. The important decision variables associated with using field data particularly WOB and the rotary speed of the drilling optimization problems are the trajectories (or the drill string for a given roller-cutter bit to minimize the histories) of (i) rotary speed of the drill bit, (ii) WOB, (iii) drilling cost for a single bit run. Iqbal (2008) presented a equivalent circulation density (ECD) and (iv) Reynolds stepwise drilling optimization procedure using real time data number function based on the circulating drilling mud through for roller-cutter bit insertion. In this methodology, the WOB the drill bit nozzle while drilling. It is also mentioned that this exponent was evaluated first for a given ROP, and then the LVWKH¿UVWVWXG\LQWKH¿HOGRIPXOWLREMHFWLYHRSWLPL]DWLRQRI optimum rotary speed of the drill string and WOB parameters RLOZHOOGULOOLQJLQYROYLQJFRQÀLFWLQJREMHFWLYHV were determined using correlations. Recently, Bahari and 2 Formulation Baradaran (2007; 2009) have proposed to minimize drilling cost for the Iranian Khangiran gas field based on the ROP 2.1 Mathematical model for drilling operation model of Bourgoyne and Young (1974). 2.1.1 Rate of penetration So far, drilling optimization studies are mostly based The drilling model proposed by Bourgoyne and Young on minimizing the cost of drilling operation. However, the (1974) is used to predict ROP while drilling. This model is optimal drilling process operation is associated with several simple and accurate involving most of the drilling parameters, important conflicting and non-commensurate objectives such as nature of formation, type of drill bit and operating that need to be optimized simultaneously in the presence drilling variables (e.g., rotary speed of the drill bit, WOB, of suitable constraints. In addition to minimize the cost ECD, Reynolds number function based on the circulating of drilling, one may achieve the highest possible rate of drilling fluids through drill bit nozzles while drilling, and penetration by minimizing the drilling time and maximizing fraction of drill bit tooth wear). According to this model, the the drilling depth simultaneously for a given degree of rate of penetration (ft/h) is given by the following nonlinear fractional wear of the drill bit tooth. The above objectives equation: DUHPXWXDOO\FRQÀLFWLQJLQQDWXUHLHLQRUGHUWRPD[LPL]H the rate of penetration by maximizing the drilling depth and minimizing the drilling time, it is difficult to minimize the G ' [SH DD ' DSSJ ' cost of drilling at the same time for given fractional drill bit S G W tooth wear. Therefore, oil well drilling operation is an ideal ZZ §· §· candidate for multi-objective optimization. ¨¸ ¨¸ GG ©¹ ©¹ Multi-objective optimization has been a highly demanding W D' SSJ D U QO S F research topic in last decade as most of the real-world decision Z §· ¨¸ making problems involve trade-offs between conflicting ©¹ objectives. Over the years, the AI-based genetic algorithm (GA) has been used as a powerful tool for the optimization DDKD 5H (1) studies for scientists and engineers. Several researchers have extended the basic algorithm GA (Goldberg, 2001) to solve Details of the estimating model constants (i.e., a to a ) are multi-objective optimization problems and the topic was 1 8 OQ Pet.Sci.(2014)11:97-110 99 as described by Bourgoyne and Young (1974) and determined conditions. In subsequent sections, w/d= W(t) is considered using non-linear multiple regression analysis. IRUVLPSOL¿FDWLRQ 2.1.2 Drill bit tooth wear 2.2 Prediction of pore pressure In addition to the ROP model, prediction of the drill bit wear while drilling is also important. Usually, a drill To determine the drilling depth and drill bit wear (i.e., bit replacement takes place if the fractional tooth wear of Eqs. (1) and (2)), it is necessary to know the variation of the drill bit is more than 75% of tooth height. A composite the pore pressure with TVD. The pore pressure depends on tooth wear equation can be obtained by considering tooth fracture pressure gradient, Poisons ratio, density, surface geometry, WOB and rotary speed of the drill bit. Thus, the porosity and porosity decline constant of the formation. instantaneous rate of tooth wear is given by the following The variation of the pore pressure with TVD is given by the equation (Bourgoyne and Young, 1974): following relation (Bourgoyne et al, 2003): Z PP I 1 §· ªº 0 .' SSJ 11 e SSJ + ¨¸ gfr «» 21PP .' 2 1 GK1 + G §· ©¹ ¬¼ P[D SSJ (2) ¨¸ P I ZZ .' GW+ W §· K 0 ©¹ 11e ¨¸ 21 P .' GG ©¹ (5) In the above equation, w/d is the weight on bit per inch In Eq. (5), the fracture pressure gradient (ppg ) usually fr of bit diameter, whereas (w/d) is the bit weight per inch of max increases with depth, whereas Poisson’s ratio (μ) declines bit diameter at which the bit tooth will fail instantaneously. with the formation depth. Other parameters, namely, H , H and H are the tooth wear parameters for a given type 1 2 3 the surface formation porosity ( ), the porosity decline of bit. Now, Eq. (2) is simplified to obtain the formation rate constant (K) and the grain density (ppg ) are almost abrasiveness constant at h = 1 by the following equation: independent of the formation depth and assumed to be invariant with depth in the present study. §· 2.3 Parameter estimation for pore pressure ¨¸ 1 G §· ©¹ W W (3) To know the variation of pore pressure with TVD, it is +E ¨¸ §·ZZ +K ©¹ essential to determine K, ppg , ppg , μ, and in Eq. (5). For g fr 0 ¨¸ GG ©¹ this, the normalized weighted square of the errors (E) between the actual pore pressure and the predicted pore pressure The abrasiveness factor IJ is numerically equal to the H is minimized at different drilling depths. The possible hours of the tooth life if the bit operates at standard conditions decision variables for this error minimization problem i.e., a bit weight of 1000 lb per inch of bit diameter with are: K, ppg , ppg , μ, and . Therefore, a single objective g fr 0 speed of 100 rpm. function optimization problem for pore pressure estimation is 2.1.3 Drill bit bearing wear formulated and written as: The instantaneous rate of bearing wear depends on the Problem 1 condition of the drill bit while drilling and the rate of bearing §· SSJ wear also increases when the bearing surfaces are damaged. p,cal ªº min ( . (SSJ ,SSJ , Z,PI , ) ¨¸ 1 gfr 0 0 ¬¼ ¨¸ The bit bearing life is assumed to vary linearly with the rotary (6a) SSJ p,d ©¹ speed of the drill bit and WOB, and the corresponding drill bit bearing wear is expressed in the following form (Bourgoyne subject to (s.t.) and Young, 1974): Bounds on the decision variables: d1%1 Z §· (4) ..dd. ¨¸ dWG W 100 4 ©¹ dd g, L g g dd In the above equation, exponent b depends upon the (6b) fr, L fr fr type of bearing and the quality of drilling fluid used. The PPddP bearing constant, IJ , is calculated from dull bit grading. IIddI Usually the drill bit tooth wearing takes place at a much 0, L 0 0 faster rate than the bearing wear. So the present study considers only drill bit wear (i.e., Eq. (2)) instead of drill Subscript d in the objective function (Eq. (6a)) describes bit bearing wear (i.e., Eq. (4)) for the optimal analysis of the desired value of the pore pressure gradient. The right hand drilling operation. Therefore, Eqs. (1) and (2) are the desired side of Eq. (6a) is multiplied by a large weighting factor, w, differential equations to predict the performance of drilling which depends on the relative magnitude of square of the operation and solved simultaneously with appropriate initial errors in the objective function (Deb, 2001). SSJ SSJ SSJ SSJ SSJ SSJ PD[ PD[ PD[ 100 Pet.Sci.(2014)11:97-110 subject to (s.t.) 2.4 Drilling optimization Model equations (i.e., Eqs. (1) and (2)) with For drilling optimization, several multi-objective Initial conditions, i.e., ' (0W' ) and KW(0 )K (9c) 0 0 optimization problems are formulated involving conflicting Bounds on the decision variables: objectives. First a two-objective optimization problem is considered where minimizing the normalized final drilling 1W () 1Wdd ()1 W () time (t /t f ref U :W ():Wdd:W() ( ) of non-dimensional drilling depth (D /D ) is the second f ref (9d) objective function. Here, t and DDUHWKH¿QDOGULOOLQJWLPH UU ()WWdd ()U W () f f cL c c and drilling depth respectively for the given single drill bit U Re (WW )dd Re ( ) ReW ( ) fL f f run, whereas t and D are the constant reference final ref ref drilling time and depth for the single bit run respectively. The Constraint: major decision variables that influence ROP during drilling operation are: (i) ECD: ȡ (t), (ii) rotary speed of the drill bit: KK (9e) N(t), (iii) WOB: W(t) and (iv) Reynolds number of function of WKHFLUFXODWLQJÀXLGWKURXJKGULOOELWQR]]OHV5H (t). N(t) and Subscript d describes the desired value of the fractional W(t KDYHGLUHFWLQÀXHQFHRQWKHUDWHRISHQHWUDWLRQZKHUHDV drill bit wear in the objective functions (Eqs. (9a) and (9b)) ȡ (t) is manipulated to maintain the stability of the wellbore and constraint (Eq. (9e)). at the bottom hole by adjusting the static mud density at the In addition to above two objectives, cost of drilling is surface. Similarly, Re is also manipulated in such a way that also an important objective which will determine whether LWZLOOKHOSWRDGMXVWGULOOLQJÀXLGSURSHUWLHVDQGGULOOLQJÀXLG the drilling operation can be carried out economically or not. hydraulics while drilling. Reynolds number function based According to Bourgoyne and Young (1974), the drilling cost U T is given by the following equation: on drill bit nozzle, Re (t), iVGH¿QHGE\ where ȝ, ȡ , f c 350P G -1 q and d are the apparent viscosity at 10,000 s , ECD, mud &&W W W circulation rate and the diameter of the drill bit nozzle are b rb c t & (10) expressed in oil field units. Knowing Re (t), one may able f ' ' to calculate the jet velocity through the drill bit nozzle, and Therefore, the third objective function may also be hence the jet impacting force or the hydraulic horse power. constructed in the form of normalized drilling cost (i.e. C / +HUHLWLVPHQWLRQHGWKDWWKHGHFLVLRQYDULDEOHVDUHQRW¿[HG C ) where C is the constant reference cost. Similarly to ref ref (or unique) values and they are all varying with drilling achieve the desired values of the drill bit fractional wear (h ), depth or drilling time. Usually, drilling is carried out till the third objective function, f , is also written in the form of WKHVSHFL¿HGIUDFWLRQDOWRRWKZHDUORVVRIWKHGULOOELWh ) is a penalty function with a large weighting factor, w , and is reached. To obtain the required drill bit fractional tooth wear written as (Deb, 2001): (h WKH¿UVWREMHFWLYHIXQFWLRQ f , is written in the form of d 1 a penalty function with a large weighing factor, w , and is written as (Deb, 2001): §· & K (11) I1W (): W , (W ), U ( )W , Re ( ) Z 1 >@ 2 ¨¸ 3c f 3 &K W §· ref ©¹ d I1W (): W , (W ), U ()W , Re () Z 1 (7) >@ ¨¸ 1c f 1 WK ref ©¹ d Therefore, considering the above three conflicting The second objective function, f , involves the objective functions (i.e., Eqs. (7), (8) and (11)), one may PD[LPL]DWLRQRIWKH¿QDOQRUPDOL]HGGULOOLQJGHSWKD /D ) f ref write the following three-objective optimization problem in for a single bit run. Similarly to achieve the desired fractional the following way: tooth wear loss (i.e., h ), the objective function (f ) is written d 2 Problem 3 in the form of penalty function with a large weighing factor, w , and is given by the following equation (Deb, 2001): §· W K min I1>@ W (: W), (W ), U ( ),W Re ( ) Z 1 ^` 1c f 1¨¸ WK §· ref ©¹ d ' K (8) I1W ():W , ()W , U ()W , Re () Z 1 >@ ¨¸ 2c f 2 (12a) ' K ref ©¹ d §· ' K max I1W (:W), (W ), U ( ),W Re ( ) Z 1 >@ ^` ¨¸ 2c f 2 7KHUHIRUHFRQVLGHULQJWKHDERYHWZRPXWXDOO\FRQÀLFWLQJ ' K ref ©¹ d objective functions (i.e., Eqs. (7) and (8)), one may write the (12b) following two-objective optimization problem: Problem 2 §· & K min I1W (: W), (W ), U ( ),W Re ( ) Z 1 >@ ^` ¨¸ 3c f 3 §· W K &K min I1W (: W), ()W , U ()W , Re () Z 1 ref ©¹ d >@ ^` ¨¸ 1c f 1 (9a) WK ref ©¹ d (12c) subject to (s.t.) §· ' K Model equations (i.e., Eqs. (1) and (2)) with max I1W (:W), (W ), U ( ),W Re ( ) Z 1 (9b) >@ ^` ¨¸ 2c f 2 Initial conditions, i.e., and (12d) ' K 'W (0 )' KW(0 )K ref ©¹ d 0 0 LVWKH¿UVWREMHFWLYHIXQFWLRQDQGWKHPD[LPL]DWLRQ Pet.Sci.(2014)11:97-110 101 Bounds on the decision variables: GHz, 504 MB RAM). The computational parameters to be used in the code are tuned until the best results are obtained. 1W () 1Wdd ()1 W () Eq. (6) is solved separately using simple GA for thirty different values of pore pressure (Bourgoyne and Young, :W ():Wdd:W() ( ) (12e) 1974). Usually the values of K, ppg and are independent of U g 0 UU ()WWdd ()U W () cL c c the formation depth and almost constant values are obtained Re (WW )dd Re ( ) ReW ( ) for first five consecutive drilling depths from top. Hence, fL f f the average optimal values of K, ppg and have been g 0 Constraint: considered for other drilling depths for parameter estimation. The average optimum values for these decision variables -5 -1 KK (12f) are found to be: K = 8.4×10 ft , ppg = 24.88 lb/gal and d g = 0.38. Optimum values of other two decision variables Penalty functions are usually added to or subtracted from i.e., fracture pressure gradient (ppg ) and Poisson’s ratio fr the objective functions for minimization of the objective (μ) are given in Table 1 for all the drilling depths. Variation function and maximization of the objective function of normalized fracture pressure (ppg = ppg /ppg ) and fr,n fr fr,ref respectively. The use of penalty functions involving ‘h’ in Poisson’s ratio with normalized depth (D = D/D ) are also n ref all the objective functions is to stop the integration of the shown in Figs. 1(a) and 1(b) respectively. A value of 18.2 lb/ state variable equations to ensure that one can obtain the gal is taken as a maximum fracture pressure (Table 1) and optimal solutions within the desired values of ‘h’. A popular used as reference ppg . Similarly, the normalized TVD (D ) fr n transformation for an objective function, f, that has to be is considered as D/20265 where the maximum drilling depth PLQLPL]HGWRRQHLQYROYLQJWKH¿WQHVVIXQFWLRQ F, that has (i.e., 20,265 ft) is considered as the reference drilling depth to be maximized, is given by (D ). It is noticed that fracture pressure gradient varies in the ref same way as the pore pressure varies with TVD. The sharp or (13) abnormal variation of ppg with D has been taken care of by fr,n n 1 I introducing the error function in the proposed equation. The This transformation does not alter the optimal solutions adjustable parameters were determined in accordance with (Deb, 2001). the methodology adopted by Tiwary and Guria (2010). The variation of fracture pressure gradient with the depth is given by the following equation: 3 Results and discussion ªº I 10 1' ' / 3.1 Pore pressure parameter estimation ^ ` nn ¬¼ (14) fr,n For the evaluation of drilling models and optimization 3.38 studies, it is required to estimate the unknown parameters in the pore pressure equation (Eq. (5)). Eq. (6) is solved using Table 1 Optimal variation of Poisson’s ratio and fracture pressure gradient -5 -1 (for K=8.4×10 ft , ppg =24.9 lb/gal and =0.38) a single-objective binary coded simple GA. To predict pore g 0 pressure with TVD, the offshore Louisiana well has been Depth Optimum values Depth Optimum values considered in the present study. The Louisiana formation ft μ ppg , lb/gal ft μ ppg , lb/gal fr fr (Bourgoyne and Young, 1974) shows characteristic abnormal 9515 0.25 12.9 12900 0.23 18.0 formation pressure. At about 11,940.0 ft depth the formation 9830 0.25 13.0 12975 0.22 18.0 pressure increases abnormally. The details of the drill data 10130 0.24 12.8 13055 0.21 18.0 along with pore pressure data are given by Bourgoyne and Young (1974). Here, the possible decision variables that 10250 0.25 12.9 13250 0.21 18.0 influence pore pressure are: porosity decline constant (K), 10390 0.24 12.8 13795 0.21 18.0 grain density (ppg ), fracture pressure gradient (ppg ), g fr Poisson’s ratio (μ) and surface porosity ( ). Upper and lower 0 10500 0.23 12.7 14010 0.21 18.0 limits of these decision variables are chosen according to 10575 0.23 12.8 14455 0.21 18.0 -5 -1 Bourgoyne et al. (2003) and are given by: 8.0×10 ft K -5 -1 10840 0.23 12.8 14695 0.21 18.0 9.0×10 ft , 20.0 lb/gal ppg 30.0 lb/gal, 8.33 lb/gal ppg 25.0 lb/gal, 0.18 μ 0.28 and 0.35 0.42. fr 0 10960 0.23 12.9 14905 0.21 18.2 The value of the weighting factor, w LV¿[HGDW for this 11060 0.23 12.9 15350 0.20 18.2 problem. The best values of computational parameters used for solving this optimization problem are: the maximum 11475 0.23 12.9 15740 0.20 18.2 number of generations = 100000, the size of population = 11775 0.24 13.6 16155 0.20 18.2 100, the length of substring = 32, the length of chromosome 11940 0.24 15.4 16325 0.20 18.2 = 32×5 = 160, the crossover probability = 0.95, the mutation probability = 0.01 and the random seed number = 0.6859. The 12070 0.22 15.6 17060 0.20 18.2 central processing unit (CPU) time required for this problem 12315 0.23 16.8 20265 0.19 18.2 is ~5.0 minutes on an Intel computer (CPU T2050 @ 1.60 SSJ HU 102 Pet.Sci.(2014)11:97-110 The calculated normalized ppg (i.e., ppg in Eq. (14)) drilling depth has been calculated using Eq. (1) with model fr fr,n and the actual normalized ppg (Bourgoyne and Young, 1974) parameters for well No. 1 with a depth range of from 9,500 to fr have been compared at different normalized drilling depths 20,000 ft of the Gulf coast area of Louisiana (i.e., a = 3.78, -3 -3 -4 and results are found to be satisfactory (Fig. 1(a)). Similarly, a = 0.17×10 , a = 0.20×10 , a = 0.43×10 , a = 0.43, a 2 3 4 5 6 the variation of Poisson’s ratio with D (Fig. 1(b)) is given = 0.21, a = 0.41 and a = 0.16). Details of the comparison n 7 8 E\WKHIROORZLQJHTXDWLRQZLWKDFRUUHODWLRQFRHI¿FLHQWPRUH between calculated and predicted ROP for the offshore than 0.9: Louisiana formation is shown in Fig. 2(a), whereas the percent error prediction for ROP with drilling depth is shown (15) in Fig. 2(b). It is observed that most of the ROP predictions P 0.180'' 0.37 0.39 nn almost matched with the actual values (except at 10,500 and To check the accuracy of pore pressure prediction, 14,000 ft depths with errors of 17 % and 15 % respectively) Eqs. (14) and (15) along with other optimum pore pressure and the calculated average absolute deviation for all depths is parameters (i.e., K, ppg and ), ROP at a different found to be only 0.23 %. g 0 1.1 0.30 (b) (a) 1.0 0.27 0.9 0.24 0.8 0.21 0.7 Estimated (Problem 1) 0.18 0.6 Predicted (Eq. (14)) 0.5 0.15 0.4 0.6 0.8 1.0 1.2 0.4 0.6 0.8 1.0 1.2 Normalized true vertical depth D/D Normalized true vertical depth D/D ref ref Variation of (a) normalized fracture pressure with normalized true vertical depth, and (b) Poisson’s ratio with normalized true vertical depth Fig. 1 50 40.0 (a) Actual (b) Calculated 20.0 0.0 -20.0 0 -40.0 8000 10500 13000 15500 18000 20500 8000 10500 13000 15500 18000 20500 True vertical depth, ft True vertical depth, ft (a) Comparison of actual rate of penetration with true vertical depth, and (b) percent error of prediction for rate of penetration with true vertical depth Fig. 2 Drilling rate, ft/h Normalized fracture pressure ppg /ppg fr fr,ref Poisson's ratio % Error in ROP prediction Pet.Sci.(2014)11:97-110 103 In this study, the rock bit with bit class (1-3 to 1-4) has been 3.2 Evaluation of the drilling model considered and IJ (life of tooth at standard conditions) has In general, the state variable equation for drilling been calculated using Eq. (3) for (i) h = 1 (i.e., complete operation is written in the form of ordinary differential wearing of the drill bit), (ii) drilling time (t ) and (iii) drill bit equations (ODEs) and is given by Eq. (16). tooth wear parameters [i.e., H , H , H and (W/d) ] for the 1 2 3 max d x offshore Louisiana formation (Bourgoyne and Young, 1974). IW xu,;x 0 x (16) Details of drilling time, drill bit tooth wear parameters and dW operating parameters (i.e., N, W, ȡ and Re ) used for drilling c f where x and u are the state variable vectors and decision simulation are given in Table 2. Bit numbers 9, 18 and 21 variable vectors respectively, and are are given by x 'K , > @ (Table 2) are chosen for drilling simulation and results for drilling depth and drill bit fractional tooth wear with drilling and u()W1 W ():W , ()W , U ()W , Re () . >@ cf time are shown in Figs. 3(a) and 3(b) respectively. This Eq. (16) is an initial value problem (IVP) and is integrated simulation procedure is quite general, and can be made for using the D02EJF subroutine (available in the NAG other drill bits. The simulation results (i.e., predicted drilling FORTRAN library) for any given u(t) and initial values of the depth and fractional wear with drilling time) for drill bit No. 9, state variables. This subroutine uses Gear’s method (Gupta, 18 and 21 are quite satisfactory and very much closer to the 2010) to integrate a set of stiff ODEs. In this code, there is a actual ROP (Bourgoyne and Young, 1974). Therefore, Eqs. (1), provision for the adjustment of error tolerance. To calculate DQG FDQEHXVHGIRU¿QGLQJWKHRSWLPDOYDOXHVRIWKH the error function in pore pressure equation (i.e., Eq. (14)), drilling variables through multi-objective optimization. the S15AEF subroutine (available in the NAG FORTRAN library) is combined with the D02EJF subroutine. Drilling 3.3 Multi-objective drilling optimization simulation is carried out for a given drill bit number with The elitist non-dominated sorting genetic algorithm, different combinations of N, W, ȡ and Re using Eq. (16). c f Table 2 Details of drilling parameters for simulation (Bourgoyne and Young, 1974) Initial conditions Operating parameters Drill bit Drilling time Drill bit teeth Normalized drilling depth Rotary speed WOB ECD Reynolds number No.* t , h life IJ , h** b H Time t D (D/20265), ft/ft N, rpm W, 1000 lb/in ȡ , lb/gal function Re n c f 7 0 0.47 113 305.80 9.50 0.96 413.70 305.80 8 0 0.49 126 10.23 9.50 0.96 14.32 10.23 9 0 0.50 129 15.23 9.60 0.83 21.43 15.23 11 0 0.51 87 4.21 9.70 0.98 12.00 4.22 12 0 0.51 78 2.53 9.70 0.98 8.75 2.54 13 0 0.53 67 3.83 9.80 0.93 27.11 6.52 15 0 0.57 77 9.29 10.30 0.89 45.36 14.22 18 0 0.58 58 4.58 11.80 0.85 22.22 4.59 21 0 0.59 67 7.10 10.30 0.98 26.61 7.10 22 0 0.60 84 5.30 15.70 0.99 13.54 5.30 23 0 0.64 85 8.53 16.70 1.15 26.43 12.11 24 0 0.65 77 0.76 16.70 1.22 1.76 0.77 25 0 0.68 81 14.79 16.80 0.27 65.60 22.26 26 0 0.69 75 2.75 16.80 0.20 10.19 2.76 28 0 0.71 64 5.28 16.90 0.75 23.42 5.28 29 0 0.74 75 3.46 17.20 0.42 22.28 7.43 30 0 0.76 85 6.62 17.00 1.29 16.42 6.63 32 0 0.80 80 12.64 17.90 0.67 63.89 24.52 34 0 0.84 65 18.19 17.60 0.75 65.58 22.40 Notes: * For a given drill bit number, the tooth wear parameters i.e., H = 1.84, H = 6, H = 0.80, and (w/d) = 8.0 are 1 2 3 max considered for calculation with bit class 1-3 to 1-4. ** Drill bit teeth life is calculated using Eq. (3). 104 Pet.Sci.(2014)11:97-110 1.2 (a) (b) 1.0 Bit No. 8 0.8 Bit No. 18 Bit No. 21 Bit No. 8 11650 0.6 Bit No. 18 9600 Bit No. 21 0.4 0.2 0.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Drilling time t, h Drilling time t, h Fig. 3 Variation of (a) drilling depth and (b) fractional rate of drill bit wear with drilling time with different bits for a single bit run NSGA-II, is used to solve multi-objective function with four time variant continuous drilling variables, i.e., optimization problems (i.e., Problems 2 and 3) involving N(t), W(t), ȡ (t), Re (t ,QWKLVSUREOHPWKHQRUPDOL]HG¿QDO c f time variant decision variables, namely, N(t), W(t), ȡ (t) drilling time (t /t ) is minimized along with the maximization c f ref and Re (t). The working principle of NSGA-II is based on of final non-dimensional drilling depth (D /D ) for given f f ref the following tasks: (i) coding of the design or decision limiting fraction of drill bit tooth wear. It is assumed that rig variables, (ii) evaluation of the objective functions and (iii) time (the sum total of rotating time, trip time during bit run improvement of objective functions using genetic operators, and connection time) is also minimized when actual drilling namely, tournament selection, single point crossover, bit time (i.e., drill bit rotating time) is minimized. The value of wise mutation and parent-daughter elitism. Details of NSGA- t and D are taken as 25.00 h and 20,265 ft respectively. ref ref II have been described by Deb and co-workers (Deb, 2001; The studies are carried out for multi-objective optimization Deb et al, 2002). An initial population of a given size is using drill bit No. 28 at a depth of 14,455 ft (Table 2) to created using a random number generation subroutine. Using find the optimal values of the drilling variables for a single genetic operators (i.e., selection, cross over and mutation), bit run. Details of the two-objective functions optimization the values of the objective functions will continue to improve problem are given in Column 2 of Table 3. Here bounds of by selecting better population of the decision variables. The the decision variables are based on exploratory drilling data subroutine D02EJF is combined with the adaptive version of of the Louisiana formation (Bourgoyne and Young, 1974). NSGA-II optimization code for multi-objective optimization For example, ECD is chosen within the drilling mud window to solve initial value problem with multiple stiff differential i.e., pore pressure gradient and fracture pressure gradient equations. To account for continuous variation of the decision of the formation. At 14,455 ft depth, the predicted fracture variables with drilling time for a single bit run, ten equally pressure gradient (Eq. (14)) and the pore pressure gradient (Eq. spaced points for each decision variable were generated (5)) are calculated as 18.2 lb/gal and 16.4 lb/gal respectively. randomly using a random number generator within its upper Therefore, bounds of the ECD (ȡ ) are chosen as 16.4-18.2 and lower limits for any drilling time interval. These equi- lb/gal. Similarly, bounds of the other decision variables are spaced decision variables are then fitted into a polynomial chosen from Table 2. Eq. (9) is solved using NSGA-II and using the E02ACF subroutine (available in the NAG Library) Fig. 4 shows the feasible optimal Pareto frontier for Problem to obtain time variant decision variables. Now interpolated 2 at the 100th generation. These solutions are clearly non- polynomials [i.e., N(t), W(t), ȡ (t), Re (t)] are incorporated in dominated (non-commensurate) to each other. It is also found c f the D02EJF subroutine for solving Eq. (16). To evaluate the that the results at the 90th generation do not differ much values of error function in the pore pressure equation (Eq. from those at the 100th generation and one can reduce the (14)), the S15AEF subroutine (available in NAG Library) is computation time easily. The best values of the computational also included with the D02EJF subroutine. Therefore, NSGA- parameters used for this problem are given in Column 2 of II code combined with three NAG subroutines (i.e., D02EJF: Table 4. The weighting factors (w and w ) for this problem 1 2 for solving IVP with multiple stiff ODEs; S15AEF: for DUH¿[HGDW . A gap is observed in the optimal Pareto plots. evaluation of error function and E02ACF: for generation of Similar gaps have also been noticed by many researchers in time variant continuous decision variables) are used to solve computing intense multi-objective optimization problems the multi-objective optimization problems. The adaptive (Kasat et al, 2002; Nayak and Gupta, 2004; Kachhap and version of NSGA-II code is free of errors and tested using Guria, 2005; Rangaiah, 2009, Guria, et al, 2009). In order standards checks (Nayak and Gupta, 2004; Kachhap and to reduce this gap in Fig. 4, optimization was carried out for Guria, 2005; Guria et al, 2005a; 2005b; Iqbal and Guria, DQGJHQHUDWLRQVDQGWKHUHZDVQRVLJQL¿FDQW 2009). effect on the optimal Pareto solution with the increase in Eqs. (9) and (11) describe the multi-objective drilling generation number. This is possibly due to the attainment of optimization problems for the offshore Louisiana formation. very close optimal values of all decision variables at the two First, Eq. (9) is solved that includes two-objective functions extreme points in the gap, i.e., points ‘B’ and ‘C’ or ‘D’ and Drilling depth D, ft; Bit No. 18 & 21 Drilling depth D, ft; Bit No. 8 Fractional loss of bit wear h, % Pet.Sci.(2014)11:97-110 105 ‘E’ in Fig. 4 (Nayak and Gupta, 2004; Kachhap and Guria, highest achievable drilling depth during drilling (i.e., point E 2005). Based on engineering judgment, a decision maker can in Fig. 4). Here, the points A and E are two limiting optimal select any preferred solution from the optimal Pareto frontier solutions for the two-objective optimization problem. We also (Fig. 4). Here, we choose five different optimal solutions select points B, C and D (Fig. 4) randomly in between A and from Fig. 4 and marked as A, B, C, D and E with increasing E on the optimal Pareto frontier and the decision maker may drilling time for a single bit run. On minimizing only f , the select any point on the basis of trade-off. The optimum values shortest achievable time for drilling will obtain and this is RIWKHGHFLVLRQYDULDEOHVFRUUHVSRQGLQJWRWKHVH¿YHIHUHQWGLI shown by point A in Fig. 4. Similarly, maximizing only f , the points (i.e., A, B, C, D and E in Fig. 4) are in shown in Figs. ¿QDOQRUPDOL]HGGULOOLQJGHSWKD ) is obtained which is the 5(a)-5(d). Table 3 Details of the multi-objective optimization problems Description Problem 2 Problem 3 Objective functions:* First Final drilling depth (D /D ) Final drilling depth (D /D ) f ref f ref Second Final drilling time (t /t ) Final drilling time (t /t ) f ref f ref Third Drilling cost (C /C ) f ref Constraint: h, % 0.75 0.75 Bounds of the decision variables: N, rpm 50-100 50-100 W, 1000 lb/in 1.00-2.00 1.00-2.00 ȡ , lb/gal 16.45-18.20 16.45-18.20 Re 0.5-1.0 0.5-1.0 Model parameters: Formation Fracture pressure gradient ppg , lb/gal Eq. (13) (Fig. 1(a)) Eq. (13) (Fig. 1(a)) fr Poisson’s ratio ȝ Eq. (14) (Fig. 1(b)) Eq. (14) (Fig. 1(b)) Grain density ppg , lb/gal 24.88 24.88 -5 -5 Porosity decline constant K, lb/gal 8.4×10 8.4×10 Surface formation porosity 0.3795 0.3795 Model constants for a depth range of 9500-20000 ft (Bourgoyne and Young, 1974) -3 -3 -4 -3 -3 -4 3.78, 0.17×10 , 0.20×10 , 0.43×10 , 3.78, 0.17×10 , 0.20× 10 , 0.43×10 , a , a , a , a , a , a , a , a 1 2 3 4 5 6 7 8 0.43, 0.21, 0.41 and 0.16 0.43, 0.21, 0.41 and 0.16 Initial conditions for ODEs (Eqs. (1) and (2)) Drilling time t, h 0.0 0.0 Drilling depth D, ft 0.0 0.0 Fractional wear of drill bit, h 0.0 0.0 Drill bit parameters Bit number 28 28 Bit class 1-3 to 1-4 1-3 to 1-4 (w/d) , 1000 lb/in 8.00 8.00 max (w/d) , 1000 lb/in 0.50 0.50 First constants for bit H Table 2 Table 2 Second constants for bit H Table 2 Table 2 Third constants for bit H Table 2 Table 2 Life of bit teeth at standard condition IJ , h 5.28 5.28 Rig parameters (Bourgoyne and Young, 1974) Bit cost C , $ ˉ 400 Rig cost C , $/h ˉ 500 Connection time t , h 1.00 Trip time t , h ˉ 6.00 Notes: * D = 20265 ft, t = 25.0 h and C = 20.0 $/ft ref ref ref 106 Pet.Sci.(2014)11:97-110 0.070 It is observed for the shortest final drilling time (i.e., point A in Fig. 4) that the optimum values of the decision D variables (Figs. 5(a) -5(d): line A) remain almost at their 0.060 highest limit. Similarly for the longest final drilling time C (i.e., point E in Fig. 4), the optimum values of the decision variables particularly N(t) and W(t) remain almost at their 0.050 lowest limit (Figs. 5(a) and 5(c): line E) whereas ȡ (t) starts from the lowest possible value at the beginning of 0.040 drilling and increases gradually with time after attaining an intermediate value at the end (Fig. 5(b): line E). Re (t) also starts increasing from the lowest possible limit and attains 0.030 the maximum possible limit at the end (Fig. 5(d): line E). )RUWKHLQWHUPHGLDWHYDOXHVRI¿QDOGULOOLQJWLPHLH3RLQW 0.020 B, C and D in the optimal Pareto front: Fig. 4), the optimum 0 5 10 15 20 25 30 values of the decision variables are also varying with time Final drilling time t , h (Fig 5(a)-5(d)). For these three points, it is interesting to note that the optimal values of ȡ (t), W(t) and Re (t) start from the Fig. 4 Optimal Pareto solution for a two-objective c f optimization problem (Problem 2, Eq. (9)) intermediate values at the beginning of drilling and gradually 19.0 A: 6.20 h B: 7.30 h (a) A: 6.20 h B: 7.30 h (b) C: 14.20 h D: 16.67 h C: 14.20 h D: 16.67 h E: 25.09 h E: 25.09 h 100.1 18.5 99.9 18.0 99.7 17.5 17.852 51.3 50.7 17.848 17.0 50.1 17.844 49.5 16.5 10 15 20 25 30 17.840 4 10162228 45 16.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Final drilling time t , h Final drilling time, t , h 2.3 1.2 A: 6.20 h B: 7.30 h (c) A: 6.20 h B: 7.30 h (d) C: 14.20 h D: 16.67 h C: 14.20 h D: 16.67 h 2.1 E: 25.09 h E: 25.09 h 1.0 1.9 2.02 1.7 1.02 2.00 0.8 1.00 1.5 1.98 0.98 1.96 1.3 0 5 10 15 20 0.6 0.96 1.1 010 20 30 0.4 0.9 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Final drilling time t , h Final drilling time t , h f f Fig. 5 Optimal variation of the decision variables i.e., (a) N(t), (b)ȡ (t), (c) W(t) and (d) Re (t) with drilling time for Problem 2 c f Normalized drilling depth D Weight on bit W, 1000 lb/in n Rotary speed of drill bit N, rpm Equivalent circulating density , lb/gal Reynolds number function R ef Normalized drilling depth Normalized drilling depth D D n n Pet.Sci.(2014)11:97-110 107 increase with drilling time. Finally, they also attain a steady drilling time, attaining a steady values at the end, which is the value at the end which is shown in Figs 5(b)-5(d) (lines: B, C lower limit of the decision variables (Fig. 7(a)). The initial and D). It is also noticed that the optimum values of W(t) and high value of N(t) will help to increase ROP and higher values Re (t) are very close to the upper limit of the corresponding of N(t) is the preferred option to reduce the drilling cost. It decision variables at the end of drilling (Figs. 5(c) and 5(d)), is also observed that the optimum values of ȡ (t) gradually whereas ȡ (t) remains at the higher intermediate value at the increase with final drilling time and attain steady values end (Fig. 5(b)) which is an indication of wellbore stability at the end with a higher intermediate value. The optimal while drilling. The scenario is different for the optimal values of Re (t) also increases with drilling time and attains values of N(t) while the optimal drilling. For the reduced a steady value at the end which is almost at the upper limit final drilling time (Fig. 4: point B), the optimum values of of the decision variable. The scenario is different for W(t) as N(t) takes an intermediate optimum value at the beginning compared to N(t), ȡ (t) and Re (t). In order to reduce the cost c f of drilling and gradually increases with time, attaining a of drilling operation (points B and C in Fig. 6), the optimum steady value at the end which is also quite close to the upper values of the decision variable takes always higher values at limit of the decision variables (Fig. 5(a): line B). With an the beginning and attains a steady value at the end which is increase in final drilling time (Fig. 4: points C and D), the at the upper limit of the decision variables (Fig. 7(c): lines B optimum values of N(t) start from the intermediate points and and C). For higher drilling cost (13.68 $/ft: Fig 6, point A), continue to decrease with time, attaining a steady value at the end which is very much closer to the lowest limit of the decision variables (Fig. 5(a): lines C and D). It is noted that the highest possible value of ȡ (t) will be preferred for faster 14 14 C: (24.5, 0.058, 13.7) C: (24.5, 0.058, 13.7) drilling to reduce the drilling time (Fig. 4 and Fig. 5(e): point E) whereas the intermediate values ȡ (t) will be preferred c B:(16, 0.057, 10.2) B:(16, 0.057, 10.2) 13 13 for the moderate rate of drilling (Fig. 4 and Fig. 5(b): except point E). Therefore, the mutual adjustment of all the decision 12 12 A:(13, 0.05, 9.9) A:(13, 0.05, 9.9) variables will help to improve ROP and will maintain well 11 11 bore stability which are very much applicable to the real-life drilling operations. 10 10 Now, a three-objective functions optimization problem is solved involving time variant continuous drilling 9 9 0.060 0.060 variables, i.e., N(t), W(t), ȡ (t) and Re (t) for given limiting 30 30 c f 0.055 0.055 25 25 fractional tooth wear of the drill bit. This problem deals with 0.050 0.050 20 20 0.045 0.045 minimization of the normalized drilling cost (C /C ), the f ref 0.040 0.040 15 15 maximization of final normalized drill depth (D ) and the 0.035 0.035 10 10 0.030 0.030 5 5 minimization of final normalized drilling time (t /t ). The f ref values of C is taken as 20.0 $/ft in the present study. Details ref of this three-objective optimization problem are given in Pareto optimal solution for three-objective optimization Fig. 6 Column 3 of Table 3 and the best values of the computational problem (Problem 3, Eq. (12)) parameters are given in Column 3 of Table 4. The weighting factors (w , w and w IRUWKLVSUREOHPDUHDOVR¿[HGDW 1 2 3 Computational parameters used for multi-objective Table 4 10 for this problem. It is also mentioned that though the optimization problems actual cost components of drilling operation may differ from Description Problem 2 Problem 3 the chosen values (Column 3 of Table 3), the optimization Number of objective functions 2 3 procedure is quite general and applicable for any values of cost data. Eq. (12) is solved using NSGA-II and Fig. 6 shows Number of decision variables 4 4 the feasible optimal Pareto surface for Problem 3 at the 100th Number of constraints 1 1 generation. It is mentioned that all the points on the optimal Number of interpolating data points for Pareto surface are non-dominated (non-commensurate) to 10 10 a decision variable each other, and the decision maker can choose any point from Length of substring l 32 32 substr this surface depending upon the trade-off involved during drilling. It is also found that the results at the 90th generation Length of chromosome l 32×40 = 1280 32×40 = 1280 chrom do not differ much from those at the 100th generation and Maximum number of generations N 100 100 gmax one can reduce the computation time easily. Similar to Number of chromosomes in the 100 100 Problem 2, any three points are selected randomly from the population N Pareto optimal surface with increasing order of drilling cost Crossover probability p 0.90 0.90 and marked as A, B and C (Fig. 6). The details of A, B and Mutation probability p 0.01 0.10 C points are also shown in Fig. 6. Variation of the optimal values of the decision variables [i.e., N(t), W(t),ȡ (t) and Random seed number 0.5789 0.5786 Re (t) are shown in Figs. 7(a)-7(d). The optimum values Computational time, min 9.50 9.85 N(t) starts with a higher value and gradually decreases with Final drilling time Final drilling time tt , h , h ff Drilling cost C , $/ft Drilling cost C , $/ft ff 108 Pet.Sci.(2014)11:97-110 19.0 A: 24.5 h, 0.058 ft/ft, 13.68 $/ft (a) (b) A: 24.5 h, 0.058 ft/ft, 13.68 $/ft B: 15.8 h, 0.057 ft/ft, 10.14 $/ft B: 15.8 h, 0.057 ft/ft, 10.14 $/ft 18.5 C: 13.01 h, 0.052 ft/ft, 9.95 $/ft C: 13.01 h, 0.052 ft/ft, 9.95 $/ft 90 ȡ 18.0 17.5 17.0 16.5 16.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Final drilling time t , h f Final drilling time t , h 2.2 1.2 A: 24.5 h, 0.058 ft/ft, 13.68 $/ft (c) (d) B: 15.8 h, 0.057 ft/ft, 10.14 $/ft 2.0 C: 13.01 h, 0.052 ft/ft, 9.95 $/ft A: 24.5 h, 0.058 ft/ft, 13.68 $/ft 1.0 1.8 B: 15 ..8 h, 0.057 ft/ft, 10.14 $/ft C: 13.01 h, 0.052 ft/ft, 9.95 $/ft 1.6 0.8 1.4 1.2 0.6 1.0 0.4 0.8 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Final drilling time t , h Final drilling time t , h f Fig. 7 Optimal variation of the decision variables i.e., (a) N(t), (b) ȡ (t), (c) W(t) and (d) Re (t) with drilling time for Problem 3 c f the optimum values of W(t) starts with a intermediate value at formulated and solved. The important objectives functions the beginning and attains a lowest possible steady value at the are: (i) maximization of the drilling depth, (ii) minimization end (Fig. 7(c), line A). The initial higher values of the W(t) of the drilling time, and (iii) minimization of the cost of is mainly to improve rate of penetration by reducing drilling drilling. The time variant decision variables present in time. It is mentioned that almost similar variation of these the multi-objective optimization problems are: equivalent decision variables are also commonly noticed in real life circulation density, rotary speed of the drill bit, weight on drill drilling operation. The above multi-objective optimization bit and Reynolds number function based on the circulating procedure is quite general and can be applied easily for fluid through the drill bit nozzles. A set of equally good HI¿FLHQWGULOOLQJRSHUDWLRQRIGHYHORSPHQWDORLOZHOOV non-dominated optimal Pareto optimal frontier is obtained for the two-objective optimization problem. Similarly, a 4 Conclusions set of equally good non-dominated optimal Pareto surfaces Multi-objective drilling operation of the Louisiana is obtained for the three-objective optimization problem. offshore field is carried out using binary coded elitist non- Depending on the trade-offs involved among the objectives, dominated sorting genetic algorithm. Bourgoyne and Young’s the decision maker may select any point from the optimal model is used to predict drilling depth and fractional drill Pareto frontier or optimal Pareto surface and corresponding bit tooth wear with drilling time. For the model based time variant decision variables for the optimal operation of optimization, the variation of pore pressure gradient with true developmental oil well drilling. The optimum values of time vertical depth is estimated by minimizing the weighted square variant decision variables are needed to be kept at the higher of the errors between actual and predicted pore pressure. values to minimize drilling time and the drilling cost whereas Several two- and three-objective optimization problems drilling depth will be maximized by keeping the optimum involving operating and economic criteria have been values decision variables at the lower values. Weight on bit W , 1000 lb/in Rotary speed of drill bit N, rpm Reynolds number function R ef Equivalent circulating density , lb/gal c Pet.Sci.(2014)11:97-110 109 f Fracture Nomenclatures g Grain a Formation strength parameter 1 L Lower bound of the decision variables a Exponent of the normal compaction trend 2 max Maximum a Under compaction exponent 3 min Minimum a Pressure differential exponent 4 n Normalized a Bit weight exponent 5 p Pore a Rotary speed exponent 6 ref Reference a Tooth wear exponent 7 t Threshold a Hydraulic exponent 8 U Upper bounds of the decision variables b Bearing exponent in Eq. (4) 0 Onitial condition/surface B Fractional bearing wear C Cost of bit, $ References C Drilling cost per foot, $/ft Ada ms N J. Drilling Engineering: A Complete Well Planning Approach. C Fixed operating cost of the rig per unit time, $/h Tulsa, Oklahoma: PennWell Publishing Company. 1985 d Drill bit diameter, in Bah ari A and Baradaran S A. Drilling cost optimization in Iranian d Bit nozzle diameter, in Khangiran Gas Field. Paper SPE 108246 presented at the D Drilling depth, ft International Oil Conference and Exhibition in Mexico, 27-30 June E Error in Eq. (6a) 2007, Veracruz, Mexico f Objective functions 1-3 Bah ari A and Baradaran S A. Drilling cost optimization in a hydrocarbon F Fitness function, Eq. (13) field by combination of comparative and mathematical methods. h Fractional tooth height worn away Petroleum Science. 2009. 6: 451-463 H , H , H Constants in Eqs. (2) and (3) 1 2 3 Bin gham M G. A new approach to interpreting rock drillability. Oil and -1 K Porosity decline constant, ft Gas Journal. 1964. 62: 80-85 l Length of chromosome Bou rgoyne A T and Young F S. A multiple regression approach to chrom optimal drilling and abnormal pressure detection. SPE Journal. l Length of substring substr 1974. 14: 371-384 N Rotary speed of drill bit, rpm Bou rgoyne A T, Millheim K K, Chenevert M E, et al. Applied Drilling N Number of chromosomes in the populatio Engineering. Richardson, TX: SPE Textbook Series. 2003 N Generation number Buc kingham E. On physical similar systems; illustrations of the use of N Maximum number of generations gmax dimensional equations. Physical Review. 1914. 4: 345-376 ppg Fracture pressure gradient, lb/gal fr Coe llo Coello C A, van Veldhuizen D A and Lamont G B. Evolutionary ppg Grain density, lb/gal Algorithms for Solving Multi-objective Problems. New York: ppg Pore pressure of the formation, lb/gal Kluwer. 2002 p Crossover probability c Deb K. Multi-objective Optimization using Evolutionary Algorithms. p Mutation probability Chichester: Wiley. 2001 Re Reynolds number function based on nozzle Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions. Evolutionary diameter of drill bit Computation. 2002. 6: 182-197 t Time, h Gol dberg D E. Genetic Algorithm in Search, Optimization & Machine t Trip time during a bit run, h Learning. New Delhi: Pearson Education Asia. 2001 t Connection time or non-rotating time, h Gra ham J W and Muench N L. Analytical determination of optimum bit t Rotating time during a bit run, h weight and rotary speed combinations. Paper SPE 1349-G presented u Decision variable vectors, Eq. (16) at SPE-AIME 34th Annual Fall Meeting, 4-7 October 1959, Dallas, Texas, USA Weight on bit per inch of bit diameter, 1000 1b/in : W G Gur ia C, Bhattacharya P K and Gupta S K. Multi-objective optimization w Weighting factors of reverse osmosis desalination units using different adaptations of 0-3 x State variable vectors, Eq. (16) the non-dominated sorting genetic algorithm (NSGA). Computers & Chemical Engineering. 2005a. 29: 1977-1995 Greek letters Gur ia C, Verma M, Mehrotra S P. et al. Multi-objective optimal synthesis ȝ Poisson’s ratio of formation -1 and design of froth flotation circuits for mineral processing using Ȟ Apparent viscosity at 10,000 s , cP the jumping gene adaptation of genetic algorithm. Industrial & Average porosity Engineering Chemistry Research. 2005b. 44: 2621-2633 Surface porosity Gur ia C, Verma M, Gupta S K. et al. Optimal synthesis of an industrial ȡ 'HQVLW\RIWKHFLUFXODWLQJGULOOLQJÀXLGOEJDO fluorspar beneficiation plant using a jumping gene adaptation of IJ Formation abrasiveness constant or life of genetic algorithm. Mineral and Metallurgical Processing. 2009. 26: bearing at standard conditions, h 187-202 IJ Formation abrasiveness constant or life of tooth H Gup ta S K. Numerical Methods for Engineers (2nd ed.). New Delhi: at standard conditions, h New Age International Publishers. 2010 Subscripts/Superscripts Han Y, Park C and Kang J M. Prediction of nonlinear production performance in water flooding project using a multi-objective cal Calculated evolutionary algorithm. Energy, Exploration & Exploitation. 2011. d Desired 110 Pet.Sci.(2014)11:97-110 29: 129-142 Nay ak A and Gupta S K. Multi-objective optimization of semi-batch Iqb al F. Drilling optimization technique-using real time parameters. copolymerization reactors using adaptations of genetic algorithm Paper SPE 114543-RU presented at Russian Oil and Gas Technical (GA). Macromol Theory Simulation. 2004. 13: 73-85 Conference and Exhibition, Moscow, Russia. 2008 Ran gaiah G P. Multi-objective Optimization: Techniques and Iqb al J and Guria C. Optimization of an operating domestic wastewater $SSOLFDWLRQVLQ&KHPLFDO(QJLQHHULQJRUOG6LQJDSRUH6FLHQWL¿F: treatment plant using elitist non-dominated sorting genetic algorithm. 2009 Chemical Engineering Research and Design. 2009. 87: 1481-1496 Ras hid K. Optimal allocation procedure for gas-lift optimization. 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A unique computer simulation model well Industrial FCC units using elitist non-dominated sorting genetic drilling: part I – the Reza drilling model. Paper SPE 15108 presented algorithm. Industrial & Engineering Chemistry Research. 2002. 41: at the SPE 56th California Regional Meeting of SPE, 2-4 April 1986, 4765-4776 Oakland, CA, USA Mai dla E E and Ohara S. Field verification of drilling models and Sin gh H K, Ray T and Sarker R. Optimum oil production planning computerized selection of drill bit, WOB, and drill string rotation. using infeasibility driven evolutionary algorithm. Evolutionary SPE Drilling Engineering. 1991. 6: 189-195 Computation. 2013. 21: 65-82 Mas seron J. Petroleum Economics (4th ed.). Paris: Editions Technip. Tiw ary P and Guria C. Effect of metal oxide catalysts on degradation of 1990 waste polystyrene in hydrogen at elevated temperature and pressure Mit chell B J. Advanced Oil Well Drilling Engineering Handbook and in benzene solution. 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Petroleum Science – Springer Journals
Published: Jan 24, 2014
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