A BILEVEL DECOMPOSITION TECHNIQUE FOR THE OPTIMAL PLANNING OF OFFSHORE PLATFORMS

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1 Bazilian Jounal of Chemical Engineeing ISSN Pinted in Bazil Vol. 23, No. 1, pp , Januay - Mach, 26 A BILEVEL DECOMPOSITION TECHNIQUE FOR THE OPTIMAL PLANNING OF OFFSHORE PLATFORMS M.C.A. Cavalho 1 and J.M. Pinto 1,2* 1 Depatment of Chemical Engineeing, Univesity of Sao Paulo, Sao Paulo - SP, 558-9, Bazil. 2 Othme Depatment of Chemical and Biological Sciences and Engineeing, Polytechnic Univesity, Booklyn - NY 1121, USA. pinto@poly.edu (Received: Januay 21, 24 ; Accepted: Septembe 9, 25) Abstact - Thee is a geat incentive fo developing systematic appoaches that effectively identify stategies fo planning oilfield complexes. This pape poposes an MILP that elies on a efomulation of the model developed by Tsabopoulou (UCL M.S. Dissetation, London, 2). Moeove, a bilevel decomposition technique is applied to the MILP. A maste poblem detemines the assignment of platfoms to wells and a planning subpoblem calculates the timing fo fixed assignments. Futhemoe, a heuistic seach pocedue that elies on the distance between platfoms and wells is applied in ode to educe the seach egion. Results show that the decomposition appoach using heuistic geneates optimal solutions fo instances of up to 5 wells and 25 platfoms in 1 discete time peiods that othewise could not be solved with a full-scale appoach. One impotant featue egading these instances is that they coespond to poblems of eal-wold dimension. Keywods: Oilfield exploation; Intege pogamming; Discete time; decomposition methods; Optimization. INTRODUCTION Thee is a geat incentive fo developing systematic appoaches that effectively identify stategies fo planning and designing oilfield complexes, due to the economic impact of the undelying decisions. On the othe hand, the development and application of optimization techniques in poblems that involve oilfield exploation epesents a challenging and complex poblem. The liteatue pesents models and solution techniques fo solving poblems in the design and planning of infastuctue in oilfields. This poblem has been initially pesented in the liteatue by Devine and Lesso (1972) that poposed an optimization model fo the development of offshoe oilfields. Accoding to Van den Heeve and Gossmann (2), in the past decisions that concened platfom capacities, scheduling of pefoations and poduction yields had been fequently made sepaately. Moeove, cetain assumptions wee made in ode to educe the equied computational effot. Anothe appoach was to assume a fixed pefoation schedule and then to detemine the poduction yield fom an LP (Linea Pogamming) model. A thid appoach was to detemine the pefoation schedule fo a fixed poduction yield fom an LP and subsequently ound the non intege solution to intege values o even to solve diectly the MILP in the simple cases. Fai (1973) poposed independent models fo calculating the numbe of poduction platfoms, thei capacities and the scheduling of well pefoation. Howeve, this appoach has lead to infeasible o suboptimal decisions since these two levels of decision wee not consideed in an integated model. Iye et al. (1998) poposed a multipeiod MILP fo the planning and scheduling of investment and opeation in offshoe oilfields. The fomulation incopoates the nonlinea behavio of the esevois, *To whom coespondence should be addessed

2 68 M.C.A. Cavalho and J.M. Pinto pessue constaints in the well suface and equipment constaints. The fomulation pesents a geneal obective function that optimizes a given economic indicato, such as the Net Pesent Value (NPV). A sequential decomposition technique is poposed to solve the poblem that elies on the aggegation of time peiods followed by successive disaggegating steps. Iye and Gossmann (1998) poposed a decomposition algoithm oiginally designed fo pocess netwok optimization that solves a design poblem in the educed space of binay vaiables to detemine the assignment of wells to platfoms. The planning model is then solved fo fixed values detemined in the design subpoblem. Tsabopoulou (2) poposed an MILP model fo the optimization of the exploation of oil and gas in a petoleum platfom. The poposed model is based on binay vaiables to detemine the existence of a given platfom and the potential connection between wells and platfoms. Otíz-Gomez et al. (22) developed multipeiod optimization models fo the poduction planning of wells in an oil esevoi. The mao decisions include the calculation of oil poduction pofiles and opeation/shut in times of the wells in each time peiod and the authos assume nonlinea timebehavio fo the well flowing pessue while calculating the oil poduction. Recently, Goel and Gossmann (24) consideed the optimal investment and opeational planning of gas field developments unde uncetainty in gas eseves. The authos showed that the poposed appoach yields solutions with significantly highe expected net pesent value than that of solutions obtained using a deteministic appoach. This pape poposes a efomulation of the MILP model of Tsabopoulou (2) that elies on a smalle numbe of binay vaiables that equies a smalle computational effot. Moeove, a bilevel decomposition technique poposed by Iye and Gossmann (1998) is applied to the efomulated model that is composed of assignment and planning subpoblems. The maste poblem detemines the assignment of platfoms to wells and the planning sub poblem that calculates the timing fo fixed assignments. With the decease in the numbe of binay vaiables and with the application of the decomposition technique, it becomes possible to solve poblems of ealistic dimension. Futhemoe, a heuistic-based constaint that limits the seach egion was developed and its impact on the optimal solution of the poblem and computational time is studied. The pape is stuctued as follows. In the following section, the poblem of planning the offshoe oilfield infastuctue is defined. In section 3, the poblem fomulation is epoduced as in Tsabopoulou (2). Section 4 pesents the poposed efomulated model (model MR) that contains a smalle numbe of binay vaiables. In section 5, the decomposition algoithm poposed by Iye and Gossmann (1998) is pesented and applied to the efomulated model (model MD). Model MD is then modified to include the heuistic pocedues to educe the seach egion in section 6. Examples ae given in sections 4 to 6 to validate the models and a detailed sensitivity analysis of the main paametes is pefomed. Finally, conclusions ae dawn on the appoach poposed in this wok. PROBLEM DEFINITION This poblem is concened with the optimal planning of offshoe oilfield infastuctue. An offshoe oilfield consists of J wells that contain oil and gas. A set of I platfoms ae equied to extact these substances fom one o moe wells. The planning decisions ae elated to the assignment of platfoms to wells in addition to the timing of extaction and poduction. Figue 1 shows the oilfield infastuctue as well as its elements. Figue 1: Poblem epesentation (Van den Heeve and Gossmann, 2) Bazilian Jounal of Chemical Engineeing

3 A Bilevel Decomposition Technique 69 Figue 1 epesents the eal-wold poblem in which seveal oilfields, esevois, wells and platfoms ae consideed. The scope of this pape is the infastuctue planning of a single oilfield that contains a set of wells. Resevois ae not explicitly consideed and in pinciple all wells can be connected to all platfoms. The cases of multiple fields and esevois will be consideed in futue wok. MATHEMATICAL MODEL The planning of infastuctue in offshoe oilfields includes discete and continuous decisions along the poect lifetime. Discete vaiables epesent the installation of platfoms and wells in each peiod. Continuous vaiables ae concened with oil and gas poduction. Based on these consideations, the model that epesents the infastuctue is a Mixed Intege Pogamming (MIP) poblem. In the case of the planning of infastuctue of petoleum fields, MINLP models have been avoided in favo of MILP o even LP models, because of the inheent difficulties of teating nonlinea constaints and in the latte case because of the combinatoial explosion that esults fom discete decisions. Despite the fact that many authos popose MINLP models that in pinciple ae moe suitable to epesent the system behavio, in this pape we ely on a linea model. The main motivation is to geneate models that ae simple and bette stuctued to solve and theefoe lage instances can be handled. Model Assumptions The following ae the main assumptions of the poposed model: (A1) Only two substances ae emoved, which ae oil and gas. (A2) The poductivity index (PI) is constant fo each well thoughout the planning hoizon. (A3) Wheneve oil is emoved fom a esevoi, its pessue deceases linealy. (A4) The pessue is the same in each well at any given time peiod. (A5) Thee is no pessue loss along the pipelines that connect the wells and the platfoms. (A6) A linea model epesents the gas-to-oil ate. (A7) The initial amounts of each substance ae known fo each well. (A8) The poduction limit fo each substance is known along the planning hoizon. (A9) The aea of the field is known and it is divided into a ectangula gid and it is possible to allocate a platfom in the cente of each ectangle. (A1) The wells ae andomly distibuted in the field. (A12) The planning time hoizon is discetized in intevals of equal length. (A13) Poduction costs and yields fo all substances ae known fo each time peiod. (A14) Inteest and inflation ates ae constant along the planning hoizon. (A15) Investment costs ae epesented by fixed paametes and ae not subect to depeciation. Thee ae petinent assumptions and those that ae consideed a elaxation fo the model. Assumptions A1, A4, A7, A8 and A13 ae petinent and the othes can be elaxed and ae discussed in the sequence. Iye et al. (1998) state that PI, a measue of the daily amount of fluids which an oil well can poduce pe unit of esevoi pessue, depends on the pemeability-thickness poduct which is obtained fom a geological map of each esevoi. Values fo PI ae obtained fom andom sampling fom a nomal distibution fo a given mean and standad deviation fo each esevoi. Accoding to this pape, PI is assumed to be constant thoughout the planning hoizon, as pesented in the second assumption. Van den Heeve and Gossmann (2) pesented a model in which the non linea behavio of the esevois is incopoated diectly in the fomulation. Based on thei assumptions, the esevois contain a substantial volume of gas so that a single linea constaint would be impecise if the pessue vaied ove a long inteval. Moeove, the gas-to-oil ate is teated as a nonlinea function of the oil emoved fom the well. Note that Iye et al. (1998) aleady consideed nonlinea behavio of the esevoi though a piecewise linea intepolation, including binay vaiables and Van den Heeve and Gossmann (2) included the nonlinea model diectly, educing the numbe of binay vaiables. Theefoe, the thid and sixth assumptions in the pesent model epesent simplifications ove pevious ones in ode to impove computational efficiency. The aea of the field is known and platfom allocation is detemined at the poect level. We adopt the assumption to allocate a platfom in the cente of the ectangle has the obective of discetizing the aea into a finite numbe of locations. Potential location of the wells is known. In the absence of eal data, thei location is andomly distibuted in the field. Infomation egading taxes and inflation ates ae time, dependent specific peiod and county dependent and can be easily adapted to the poblem instance. Dilling and connection costs ae consideed fixed paametes. Iye and Gossmann (1998) consideed that dilling and poduction costs have fixed and vaiable components that depend on the Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

4 7 M.C.A. Cavalho and J.M. Pinto capacity of the platfoms. The authos conside that the maximum capacity is equal to the lagest value obtained in all time peiods. In the model developed by Tsabopoulou (2) platfom capacity was not consideed and theefoe vaiable costs ae not included. In summay, assumptions A2, A3, A5, A6 and A15 ae those that could be teated in a moe ealistic way by intoducing non lineaities to the model. Common methods of capital budgeting include net pesent value (NPV), i.e. the pesent value of cash inflow is subtacted fom the pesent value of cash outflows. NPV compaes the value of a cuency unit today vesus the value of that same cuency unit in the futue afte taking inflation and etun into account. If the NPV of a pospective poect is positive then it should be accepted, othewise the poect pobably should be eected because cash flows ae negative. In this context, the obective is to maximize the NPV. The MILP model oiginally poposed by Tsabopoulou (2) maximizes NPV and it is epoduced below (also denoted as Model MO). The obective function in Equation 1 includes the evenues of oil and gas defined as GAS and OIL, educed by the dilling and connection costs, DR and CON, espectively. Max NPV=GAS+OIL-DR-CON (1) s.t. o,,t t t (2) t OIL = F (APO PCO) D g,,t t t (3) t GAS = F (APG PCG) D Equations 2 and 3 ae elated to the evenues fom oil and gas that depend on annual oil and gas pices at time peiod t, APO t and APG t, espectively. These pices ae subtacted fom thei poduction costs, PCO and PCG. Moeove, evenues depend on depeciation D t. The geneal equation fo depeciation D t is the following: 1+ INFLATION D= t 1+ INTEREST t 1 t (4) Equations 5 and 6 ae based on assumption A15 and ae elated to the dilling and connection costs, espectively. i i, (5) i DR = (1M + 1 X ) 1 i, i, (6) i CON = COST X The cost depends diectly on the assignment of the well to the platfom whee connection cost COST i, between wells and platfoms is the same as the one mentioned by Devine and Lesso (1972): COST = WD i, H WD H +5 ( 1.5) 2 i 2 i, WD i, (7) whee WD is the depth of well and H i, is the hoizontal distance between well to be connected to platfom i. The hoizontal distance shown in Equation 7 is a function of platfom and well co-odinates. Its coesponding equation may be witten as: 2 2 ( ) H = (PX WX ) + (PY WY ) i, (8) i, i i Pocess conditions ae assumed to have linea behavio accoding to assumptions A3 and A6 and ae epesented as follows. CUM = CUM + F s,t (9) t s,t s,t 1 s,,t 6 P = CUM t (1) Equation 9 states that the cumulative poduction of each substance (oil/gas) is the same as the cumulative poduction in the pevious time peiod inceased by an amount equal to the flow fom all wells at the pesent time. Equation 1 states that the initial pessue of the esevoi is 1 ba and that it deceases linealy with accumulated poduction (in baels). FMAXo,,t = PI P t,t (11) 6 FMAXg,,t = PI ( CUM o,t) (12),t F FMAX s,,t (13) s,,t s,,t o,t Fs,,t INVAL s, s, t 1 2 (14) Equations 11 and 12 ae elated to the maximum flow of poduction of the oil and gas in Bazilian Jounal of Chemical Engineeing

5 A Bilevel Decomposition Technique 71 baels, espectively. Equation 13 states that the flow of each substance fom each well should not exceed the maximum poduction limits. Equation 14 enfoces that the flow of all substances thoughout the time hoizon should not exceed thei initial amounts. a = a + x,t (15),t,t 1 i,,t i Equation 15 states that a well is opened only once and emains open thoughout the whole time hoizon. Note also that a, = fo evey well ; in othe wods, the well will eventually be made available duing the planning hoizon. F FOMAX a,t (16) o,,t g,,t,t F FGMAX a,t (17) o,,t,t,t F FOMIN a,t (18) F FGMIN a,t (19) g,,t,t Constaints 16 to 19 state that the oil and gas flow should not exceed uppe and lowe bounds. Futhemoe, note that Equations 16 and 17 set the oil and gas flow ates to zeo in case a well is not made available. Logical Constaints 2 to 24 elate the decision vaiables fom the model. X = x i, (2) i, t i,,t Yi,t M i i t x (21) Y i,,t (22) i,,t i,t t xi,,t 1 i t xi,,t Mi i, t (23) (24) Equation 2 states that a well is connected to a platfom only if it has been connected to the same platfom at one time peiod duing the whole time hoizon. Equation 21 enfoces that evey platfom is installed at most once within the whole time hoizon. Equation 22 states that if a well is connected to a platfom duing the whole time peiod, the coesponding platfom has to be installed. Equation 23 enfoces that a well is connected to a platfom at most once. Equation 24 states that a well is connected to a platfom only if the same platfom was allocated. REFORMULATED MODEL Poblem MR coesponds to a efomulation of the model poposed by Tsabopoulou (2) and pesented in the pevious section. The main diffeence between both models elies on the epesentation of the binay decision vaiables. Model MR Tsabopoulou (2) consideed five sets of binay vaiables. The fist set assigns wells to platfoms (X i, ), the second and thid epesent the selection and the timing of platfoms (M i and Y i,t ), fouth the availability of wells (a,t ) and the last one elates wells to platfoms at evey time peiod (x i,,t ). The efomulated model (MR) contains only the last thee sets of vaiables, which is sufficient to model the discete decisions of the poblem. MR: Max NPV=GAS+OIL-DR-CON (1) s.t. constaints (2) and (3) (9) to (17) (23) and (24) i i,,t (25) i t DR = (1 M + 1 x ) 1 i, i,,t (26) i t CON = COST x Fs,,t s,,t (27) Note that Constaints 18 and 19 wee eliminated because the lowe bounds fo the flow ates (FOMIN and FGMIN) ae set to zeo. On the othe hand, non negativity constaints fo the flow ates ae imposed in the model in 27. Model MO uses vaiable X i, to connect platfom i to well and x i,,t to connect platfom i to well at time t. Futhemoe, vaiables Y i,t denote the time t at which platfom i is installed. All these decisions can be epesented by x i,,t. Theefoe, Equations 2 to 22 ae unnecessay and Equations 5 and 6 ae tansfomed into Equations 25 and 26, espectively. Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

6 72 M.C.A. Cavalho and J.M. Pinto RESULTS In this section we pesent in detail a case study as the one intoduced by Tsabopoulou (2) that povides a compaison between MO and the poposed model MR. Fo this case, 16 platfoms and 3 wells ae consideed fo a hoizon of 1 yeas that is divided into equal time peiods of 1 yea each. In this poblem, a ectangula oilfield of 1, ft by 15, ft is consideed (Figue 2). Uppe poduction limits of oil and gas in each well ae 1,25, ft 3 and 875, ft 3, espectively. Inteest and inflation ates wee set by Tsabopoulou (2) to 15% and 3%, espectively. Data egading poductivity indexes (PI), initial amounts of substances (oil and gas), the coodinates in the field, and depth (WD) fom each well ae given in Table 1. WX (ft) WY (ft) Figue 2: Configuation of field Table 1: Data fo each well WD (ft) PI INVAL bael (1 5 bael/yea) y.ba Oil Gas Bazilian Jounal of Chemical Engineeing

7 A Bilevel Decomposition Technique 73 The poblem is solved to illustate the pefomance of the models and of the solution stategy. The MILP poblems wee modeled using GAMS (Booke et al., 1998) and solved in full space using the LP-based banch and bound method implemented in the CPLEX solve (ILOG, 1999). The efomulated model (MR) pesented bette computational pefomance with espect to the oiginal model (MO) poposed by Tsabopoulou (2), as shown in Table 2 that pesents the CPU times obtained fo a poblem with 16 platfoms as a function of the numbe of wells (). Inteestingly, the integality gap is the same fo both models and inceases with poblem size. Table 3 pesents the sizes of MO and MR, such as the numbe of single equations (SE), the numbe of continuous vaiables (SV) and the numbe of discete vaiables (DV) fo seveal numbes of wells () and 16 platfoms. Note fom Table 3 that thee is a linea incease in the numbe of equations as well as in the binay and continuous vaiables with the incement of the numbe of wells. It is impotant to note that the two models in pinciple might not have the same integality gap because they ae based on diffeent fomulations (Williams, 1999). In that espect, modeling is lagely an at that has a lage impact in mixed-intege pogamming (Biegle et al., 1997) and computational expeiments ae necessay to test and compae fomulations. Solution pefomance depends on seveal factos such as model size (constaints, binay vaiables and continuous vaiables) and model fomulation. In this paticula case, the fome played the most significant ole in educing computational effot. Figue 3 illustates the computational time fo MO and MR fo diffeent numbes of wells and 16 platfoms and shows that the latte is smalle than the one fo MO unde any configuation. Howeve, the computational effot pesents a non-linea behavio with the numbe of wells. Table 2: Computational pefomance of the models CPU time (s) Gap MO MR (%) * * - * No intege solution obtained afte 18, CPU s. Table 3: Dimensions of MO and MR MO MR SE SV DV SE SV DV CPU (s) MO MR Figue 3: CPU times fo MO and MR Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

8 74 M.C.A. Cavalho and J.M. Pinto Sensitivity Analysis A sensitivity analysis of the main model paametes was pefomed. Figues 4 to 6 show the elative change of the obective function (NPV) with espect to the base case the paametes in the -2% to +2% ange. Figue 4 shows the esults fo the main paametes that impact the obective function that ae INTEREST, INFLATION, PCO and PCG. Note that the fist two paametes diectly affect depeciation. The ones that most significantly impact NPV ae PCO and INTEREST. Figue 5 pesents the influence of FOMAX, FGMAX and INVAL. It can be seen that paamete INVAL has a small impact on the obective function, wheeas thee is no sensitivity of NPV on the two othes because constaints (16) to (19) ae not active at any time peiod. The main pupose of Figue 6 is to veify the influence of the paametes that take pat in the pessue constaint (Equation 1). These paametes ae α and β denote the intecept and the slope of 6 Equation 1 (values 1 and 8. 1 in the base case), espectively. Hence, it can be obseved fom Figue 6 that the initial pessue has a stonge influence than the decease ate. % paamete vaiation INTEREST INFLATION PCO PCG % NPV Figue 4: Sensitivity of the obective function paametes % paamete vaiation INVAL FOMAX and FGMAX % NPV Figue 5: Sensitivity of FOMAX, FGMAX and INVAL % paamete vaiation % NPV Figue 6: Sensitivity of the pessue constaints α β BILEVEL DECOMPOSITION APPROACH Fom the esults of the pevious section it becomes clea that neithe model can efficiently solve poblems of lage sizes if MILP solves wee to tackle them in full-scale. Theefoe, a decomposition appoach is applied to model MR. Iye and Gossmann (1998) poposed a two-level decomposition appoach fo the planning of pocess netwoks. Van de Heeve and Gossmann (2) then applied this technique to an oilfield infastuctue-planning model. In this section, a simila appoach is applied to the efomulated model MR. The esulting model is denoted as MD that is decomposed into two subpoblems: the maste subpoblem that solves a model that assigns platfoms to wells (poblem AP) and the timing subpoblem (poblem TP). The latte elies on the assignments that ae obtained in the maste subpoblem and decides on when to install the platfoms. The decomposition algoithm as applied to model MR can be seen in Figue 7. In Figue 7, design cuts coespond to Constaints 35 to 37 that ae descibed in item 5.1. The poposed technique is simila to the one poposed by Van den Heeve and Gossmann (2), which howeve have consideed non convex nonlineaities in the sub-poblem and theefoe could not guaantee global solutions. Bazilian Jounal of Chemical Engineeing

9 A Bilevel Decomposition Technique 75 Figue 7: Bilevel decomposition algoithm. Model MD The model is solved iteatively such that the two MILP sub-poblems AP and TP ae optimized in each iteation. The assignment poblem (AP) is defined as follows: max NPV=GAS+OIL-DR-CON (1) s.t. constaints (2) and (3) (5) and (6) (9) to (14) and 27 A = X (28) o,,t i i, F FOMAX A,t (29) g,,t F FGMAX A,t (3) Xi, 1 i X i i (31) M i, (32) In (28), vaiable A is assigned to one when a well is opened. Note that the availability of the well is no longe associated to time and that the assignment vaiable X i, oiginally defined by Tsabopoulou (2) is intoduced. In (29) and (3), the flow ates of both oil and gas should neve be above specific limits FOMAX and FGMAX, espectively. In (31) it is clea that both wells and platfoms ae connected only once within the hoizon. Futhemoe, Equation 32 states that one well is connected to a platfom only if this was installed. The solution of AP povides values fo X i,. If this vaiable is fixed, denoted by X i,, a feasible solution fo TP is a feasible solution fo MR and geneates a lowe bound fo this poblem. The timing poblem (TP R ), at iteation R, is defined as follows: max NPV=GAS+OIL-DR-CON (1) s.t. constaints (2) and (3) (9) to (17) (23) to (27) x X i,,t (33) a i,,t,t i, A,t (34) Similaly to Iye and Gossmann (1998), Constaints 33 and 34 select a subset of assignments fo the planning poblem. The following ae the constaints (design cuts) used in the AP model in the algoithm to avoid subsets and supesets that would esult in suboptimal solutions: Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

10 76 M.C.A. Cavalho and J.M. Pinto (n1,n2) Z1 n1,n2 + i, 1 X X Z (i,) Z,=1...R (n1,n2) Z 1 n1,n2 i, X + X 1 (i,) Z, =1...R (i,) M =1...R whee i, i, (i,) N X X M 1 (35) (36) (37) { } { } { } M = (i, ) / X = 1 fo configuation in iteation N = (i, ) / X = fo configuation in iteation 1 i, Z = (i,)/x = 1 { } i, Z = (i,)/x = Similaly to Iye and Gossmann (1998), Equation 35 states that if in any solution all the X i, vaiables in any set Z 1 ae 1, then all emaining vaiables must be zeo in ode to pevent a supeset of Z 1 fom enteing the solution of AP. Equation 36 shows cuts fo pecluding subsets of Z 1. Equation 37 has the effect of establishing the basis fo deiving intege cuts on supesets and subsets of the configuations pedicted by the assignment poblem. This popety of supesets and subsets is the basis fo deiving the intege cuts. Note that the design cuts (35) to (37) accumulate along the iteations. logical (binay) vaiables of AP X i and A and those of TP, given by x i,t and a,t. The vaiables that define the connection of platfom i to well ae elated in (2). The vaiables that define the availability of wells ae elated in (38). X = x i, (2) i, t i,,t ( ) A = a a,t,t 1 t (38) By definition, A is one when the well is made available duing any time peiod within the time hoizon. This is veified by examining vaiables a,t at evey two consecutive time peiods. If the well is made available at time t, then a,t - a,t-1 = 1; othewise, the diffeence is zeo (the well is not available o it was made available at a pevious peiod, in which case a,t = a,t-1 = 1). Regading the constaints of AP that do not belong to TP, Equations (5) and (6) ae obtained by substituting (2) into (25) and (26), espectively. Constaint (28) can be obtained as follows. Conside equation (15): a = a + x,t (15),t,t 1 i,,t i Reaanging (15) and summing ove t: ( a,t a,t 1) = x i,,t t t i Using (38) and (2) yields Equation (28): (39) A = X (28) i i, Constaints (29) and (3) epesent elaxations of constaints (16) and (17), espectively; this can be shown by the elationship between A and a,t given in (34). Moeove, constaints (31) and (32) ae obtained by eplacing (2) into (23) and (24), espectively. Bounding Popeties of DP A vey impotant popety of the decomposition stategy is that AP epesents a igoous uppe bounding poblem to TP. This can be veified by compaing the feasible egion of the two poblems. Note that all constaints of AP ae also pesent in TP, with exception of (5), (6), and (28) to (32). Fistly, it is impotant to establish the elationship between the Results Fom Table 2 it is clea that models MO and MR ae unable to solve poblems with moe than 35 wells, despite a elatively small integality gap veified fo the smalle instances. Nevetheless, when MR is subect to the decomposition stategy poposed in the pevious section (denoted as MD), the computational gain is emakable. The CPU Bazilian Jounal of Chemical Engineeing

11 A Bilevel Decomposition Technique 77 times obtained fo a poblem with 16 platfoms as a function of ae compaed to those fom MR in Figue 8. Figue 8 also illustates the computational time fo MD fo diffeent numbes of wells and platfoms, anging fom 16 and 25. Table 4 pesents the coesponding sizes of poblem MD, fo seveal values of the numbe of wells. SV and DV ae maintained at each iteation, wheeas thee is an aveage incease of 2% in the numbe of equations fom iteation 1 to 2, due to the cut geneation step. It can be seen fom Table 4 that the eduction in the numbe of discete vaiables (DV) in MD is not significant with espect to MR. Howeve the intoduction of Constaints (33) and (34) geatly educes the seach space and theefoe the computational effot. The optimal values obtained with MO, MR and MD ae the same fo all cases and only 3 subpoblems ae equied fo MD fo all instances. The well-platfom assignments obtained fo MD ae given in Table 5. Note that, besides the obective function value, the decision vaiables ae expessed by platfom (wells). Fo the sake of illustation, ten iteations of the algoithm ae shown in Table 5. It is impotant to note howeve that the algoithm conveges in a single iteation. Note that Constaints 35 to 37 do not allow the epetition of assignments neithe the geneation of sub and supesets. In this sense, thee is no significant change in the allocation obtained in AP in consecutive iteations. Fo instance fom the fist to the second iteation, the only modification is the allocation of well 9 to platfom 11 in place of the assignment of well 9 to platfom 1. Note also fom Table 5 that all platfoms ae installed and connected to the wells in the fist time peiod, since no investment constaints ae imposed in the model. Theefoe assumption A15 does not epesent any simplification to the model. Results in Table 5 show that thee is a moe significant decease in NPV in the fist fou iteations. Finally, 29 allocations ae made fo all iteations. CPU (s) MR, I=16 MD, I=16 MD, I= Figue 8: Computational pefomance fo MR and MD Table 4: Size of poblem MD Sub poblem 1 st iteation SE SV DV CPU (s) NPV AP TP AP TP AP TP AP TP AP TP AP TP AP TP Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

12 78 M.C.A. Cavalho and J.M. Pinto Table 5: Assignments fo 1 iteations of MD Iteations () NPV (1 8 ) allocation X i, (i X i, =1) 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,8,22,24,29,3); 1(2,5,9,2); 11(25,28); 13(18,27); 15(17,21); 16(16) 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,8,22,24,29,3); 1(2,5,2); 11(9,25,28); 13(18,27); 15(17,21); 16(16) 1,26; 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,8,22,24,29,3); 1(2,5,9,2); 11(25,28); 13(18,27); 15(17,21) 1,26; 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,8,22,24,29,3); 1(2,5,2); 11(9,25,28); 13(18,27); 15(17,21) 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,22,24,29,3); 1(2,5,9,2); 11(25,28); 13(8,18,27); 15(17,21); 16(16) 2(3,6,7,11,14,23); 3(12,13,15,19); 8(4,1); 9(1,8,22,24,29,3); 1(2,5,9,2); 11(25,28); 13(18,27); 15(17,21); 16(16) 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,22,24,29,3); 1(2,5,2); 11(9,25,28); 13(8,18,27); 15(17,21); 16(16) 2(3,6,7,11,14,23); 3(12,13,15,19); 8(4,1); 9(1,8,22,24,29,3); 1(2,5,2); 11(9,25,28); 13(18,27); 15(17,21); 16(16) 1,26; 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,8,22,24,29,3); 1(2,5,2); 11(25,28); 13(18,27); 15(17,21); 16(16) 2(3,6,7,11,14,23); 3(12,13,15,19); 4(4,1); 9(1,8,22,24,29,3); 1(2,5,9,2); 11(25,28); 13(18,27); 15(17,21); 16(16) DECOMPOSITION APPROACH USING HEURISTIC Fom Figue 9 it becomes clea that most of the computational effot lies on the solution of AP due to the lage combinatoial aspect that esults fom allocating wells to platfoms. As solving AP is intinsically elated with the well-platfom connection, it becomes inteesting to limit some of the possible connections. Accoding to Gimmett and Stazman (1988), it is common the use of the constaint that enfoces a maximum hoizontal distance (adius) that a well can be dilled fom a fixed suface location. This constaint is impotant because it limits the numbe of wells that can be diectionally dilled fom a platfom. Such type of constaint can be included in the mathematical pogamming fomulation of the location-allocation poblem, as illustated in Figue 1. Model MH Models MO, MR and MD conside all associations between wells and platfoms. Howeve, Equation 7 shows connection costs that ae diectly elated to the hoizontal distance between well and platfom. Figue 1 illustates how the wells and platfoms ae distibuted, as suggested by assumptions A9 and A1. The aea of the field is divided in N 2 smalle ectangles of equal size and in each of them the numbe of potential platfoms allocated to each well at evey time peiod. A constaint that assigns zeo values to the X i, vaiable when the distance is lage than the smallest distance of the smalle ectangles of the field is added to model MD. So, MD, now denoted MH, has a smalle numbe of possible associations between wells and platfoms, and theefoe may be able to solve the poblem that includes heuistics with lowe computational effot. In ode to evaluate the impact of the heuistics in the model, two limiting values of association between wells and platfoms wee tested. The adius is detemined by the hoizontal distance that a well can be dilled fom a fixed suface and two connection limits ae defined as follows: fo MH1 the maximum hoizontal distance is given by the maximum between LY/N and LX/N, wheeas fo MH2, the adius is defined as the one that connects the cente to the vetices of the small ectangle (Figue 1). Bazilian Jounal of Chemical Engineeing

13 A Bilevel Decomposition Technique % CPU (AP) Figue 9: Pecentage of computational time to solve AP Figue 1: Configuation of field with heuistic Results Values of LX and LY epesent the dimensions of the field, and oiginally admit values of 15, ft and 1, ft, espectively. The value of N as well as the size of the field wee inceased with the obective of analyzing the effect of poblem dimension in the computational effot. Values ae shown in Table 6. Figues 11a and 11b illustate esults fo MD and MH1 fo diffeent numbes of wells and platfoms. Attibutions well-platfom along time fo MH1, consideing 3 wells and 16 platfoms ae the same as the ones fo model MD as well as the value of the obective function found. Howeve, CPU time was 2.4 seconds that is 35% smalle than that equested by MD and 6% smalle than equied fom MO. Note also that in Figue 11b only the case with 1 wells could be solved when 25 potential platfoms ae defined. Figues 12a and 12b illustate the impact of the heuistic on the models MILP. The same value fo the obective function was found in all models. Howeve, model MH1 equies a lage computational time than model MH2. This esult should be expected because essentially the adius detemines the seach aea. In this sense, the heuistics that uses a smalle adius tends to obtain esults with smalle computational effot. As the adius used in MH1 and MH2 ae simila, thei computational time is also simila. The adius utilized in MH1 was chosen in ode to cove the smallest ectangle. In othe wods, to allow connection of the platfom location in the cente of the ectangle to all potential wells in this ectangle. The adius utilized in MH2 was chosen in ode to evaluate the seach in elation the MH1 and to veify how the adius could influence the computational time. Table 6: Size of the field N WP LX (ft) LY (ft) , 1, ,438 15, ,75 22, ,938 3, , 4, Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

14 8 M.C.A. Cavalho and J.M. Pinto CPU (s) CPU (s) MD - I=16 MH1 - I= (a) (b) Figue 11: Computational pefomance fo MD and MH1 MD - I=16 MH1 - I=16 MH2 - I= (a) (b) Figue 12: Computational pefomance fo MH1 and MH2 CPU (s) CPU (s) MD - I=25 MH1- I= MH2 - I=25 MH1 - I= Accoding to Figues 11b and 12b, the model cannot be solved to global optimality fo 2 wells and 25 platfoms. Seveal values of the elative optimality citeion (OPTCR) wee used in the ange of 1 to 1% in model MH2. OPTCR is defined as (best estimate-best intege)/ best estimate, in which "best intege" is the best solution that satisfies all intege equiements found so fa and "best estimate" povides a bound fo the optimal intege solution. Table 7 and Figue 13 show the esults and compae the obective function and computational time fo each citeion. Despite diffeent values of OPTCR, the esulting obective function value is the same. In ode to evaluate the impact of the optimality citeion fo anothe instance of the poblem, esults wee optimized initially fo the case in which an optimal solution was obtained (1 wells and 25 platfoms). Results ae epesented by Figue 14 (also shown in Table 7). Note that the computational time pesents a significant incease when the optimally citeion is smalle than 1%, and the coesponding obective function pesented a small incease. The computational time to non null optimality citeia is smalle fo all cases, except fo 5 wells. Fo 3 wells and optimality citeion 1%, the decease in CPU is 95% and epesents a eduction in the obective function of only.5% in elation to the optimal solution. Table 7: Computational time fo 25 platfoms OPTCR Obective Function CPU (s) Bazilian Jounal of Chemical Engineeing

15 CPU (s) A Bilevel Decomposition Technique OPTCR (%) Figue 13: Computational time and obective function fo 25 platfoms and 2 wells NPV ($x1e6) CPU (s) OPTCR (%) Figue 14: Computational time and obective function fo 25 platfoms and 1 wells NPV ($x1e6) CONCLUSIONS This pape addessed the long tem planning of the oilfield infastuctue. Fistly, we poposed a efomulated MILP that pesents a significant eduction in the numbe of discete vaiables fo the same elaxation gap with espect to the model developed by Tsabopoulou (2). Moeove, a bilevel decomposition appoach that elies on the disaggegation of the assignment and timing decisions in analogy to the one poposed by Iye and Gossmann (1998) has been pesented. Results show that computational pefomance is geatly impoved, wheeas global optimality is guaanteed. Poblems of 25 platfoms and 4 wells ae efficiently solved fo a 1-yea hoizon. Finally, heuistics that limit the assignment of platfoms to wells wee poposed. Results show that gains of up to 86% in CPU time wee obtained with the addition of the heuistic ule without compomising the solution quality. ACKNOWLEDGMENT The authos acknowledge financial suppot fom CAPES (Bazil). NOMECLATURE Indices g gas i platfom well o oil s substance (gas o oil) t time peiod Paametes APO t annual oil pice at time peiod t APG t annual gas pice at time peiod t COST i, cost between wells and platfoms D t depeciation at time peiod t H i, hoizontal distance between well to be connected to platfom i FGMAX uppe bound of gas flow FGMIN lowe bound of gas flow FOMAX uppe bound of oil flow FOMIN lowe bound of oil flow INFLATION inflation ate INTEREST inteest ate INVAL s, initial value fo substance s in Bazilian Jounal of Chemical Engineeing Vol. 23, No. 1, pp , Januay - Mach, 26

16 82 M.C.A. Cavalho and J.M. Pinto LX LY N PCG PCO PI PX i PY i Q s,t WD WX WY Continuous Vaiables A CON CUM s,t DR F s,,t FMAX s,,t GAS NPV OIL P t Binay Vaiables a,t M i x i,,t X i, Y i,t well x-dimension of the oilfield y-dimension of the oilfield numbe of gids in which each distance is divided (N 2 ectangles ae fomed) oveall numbe of wells annual poduction costs fo gas annual poduction costs fo oil poductivity index fo well x co-odinate of platfom i y co-odinate of platfom i uppe poduction limit fo substance s at time peiod t depth of well x co-odinate of well y co-odinate of well availability of well connection cost cumulative poduction of substance s up to time peiod t oveall dilling cost flow ate of substance s fom well duing time peiod t maximum flow of substance s fom well at time peiod t evenues of gas obective function vaiable evenues of oil pessue of all wells at time peiod t availability of well at time peiod t existence of platfom i connection of platfom i to well at time peiod t connection of platfom i to well time peiod t at which platfom i is installed REFERENCES Biegle, L.T., Gossmann, I.E. and Westebeg, A.W. (1997). Systematic methods of chemical pocess design. Pentice Hall, Uppe Saddle Rive, NJ. Booke, A., Kendick, D. and Meeaus, A. (1998). GAMS: A Use s Guide; GAMS Copoation; The Scientific Pess: San Fancisco, CA. Devine, M.D. and Lesso, W.G. (1972). Models fo the Minimum Cost Development of Offshoe Oil Fields. Manage. Sci., Vol. 18, N o. 8, B378-B387. Fai, L.C. (1973). Economic Optimization of Offshoe Oilfield Development. PhD Dissetation, Univesity of Oklahoma, Tulsa, OK. Goel, V. and Gossmann, I.E. (24). A Stochastic Pogamming Appoach to Planning of Offshoe Gas Field Developments unde Uncetainty in Reseves. Comput. Chem. Engng, Vol. 28, Gimmett, T.T. and Statzman, R.A. (1988). Optimization of Offshoe Field Development to Minimize Investment, SPE Dilling Engineeing, Vol. 3, Ilog (1999). Ilog Cplex (6.5) Use s Manual. Ilog Cop. Gentilly, Fance. Iye, R.R. and Gossmann, I.E. (1998). A Bilevel Decomposition Algoithm fo Long-Range Planning of Pocess Netwoks. Ind. Eng. Chem. Res., Vol. 37, Iye, R.R., Gossmann, I.E., Vasanthaaan, S. and Cullick, A.S. (1998). Optimal Planning and Scheduling of Offshoe Oil Field Infastuctue Investment and Opeations. Ind. Eng. Chem. Res., Vol. 37, Otíz-Gomez, A.; Rico-Ramiez, V.; Henández- Casto, S. (22). Mixed-intege Multipeiod Model fo the Planning of Oilfield Poduction. Comput. Chem. Engng, Vol. 26, , 22. Tsabopoulou, C. (2). Optimisation of Oil Facilities and Oil Poduction, M.Sc. Dissetation, Univesity College London (UCL), London, UK. Van den Heeve, S., Gossmann, I.E., Vasanthaaan, S. and Edwads, K. (2). Integating Complex Economic Obectives with the Design and Planning of Offshoe Oilfield Infastuctues. Comput. Chem. Engng, Vol. 24, Van den Heeve, S. and Gossmann, I.E. (2). An Iteative Aggegation / Disaggegation Appoach fo the Solution of a Mixed-Intege Nonlinea Oilfield Infastuctue Planning Model. Ind. Eng. Chem. Res, Vol. 39, Williams, H.P. (1999). Model Building in Mathematical Pogamming. 4th edition, John Wiley & Sons, Chicheste, UK. Bazilian Jounal of Chemical Engineeing