1-s2.0-S0920410515000704-main

Modeling of ow splitting for production optimization in offshoregas-lifted oil elds: Simulation validation and applications

Thiago Lima Silva a, Eduardo Camponogara a,n, Alex Furtado Teixeira b, Snjezana Sunjerga c

a Department of Automation and Systems Engineering, Federal University of Santa Catarina, Cx.P. 476, Florianpolis, SC 88040-900, Brazilb Petrobras Research Center, Rio de Janeiro, RJ 21949-900, Brazilc Petroleum Engineering Reservoir Analysts, 7048 Trondheim, Norway

a r t i c l e i n f o

Article history:Received 12 July 2014Accepted 8 February 2015Available online 17 February 2015

Keywords:daily production optimizationgas-liftow splittingmixed-integer linear programming

a b s t r a c t

In modern offshore oil elds, wells can be equipped with routing valves to direct their production tomultiple manifold headers, a strategy that is routinely adopted in practice either to provide resilience toequipment failure or to improve production. However, the existing models for production optimizationdo not account for splitting of ows and therefore require the wells to be connected to a single header. Tothis end, this work develops a nonlinear model of ow splitting that reproduces the complex behaviorobserved in multiphase-ow simulation. This model is further approximated with multidimensionalpiecewise-linear functions to a desired degree of accuracy with respect to simulated behavior. Thesepiecewise-linear functions enable the development of a Mixed-Integer Linear Programming (MILP)formulation for production optimization, which decides between single and multiple routing of wells toheaders. The effectiveness of this MILP formulation is assessed in a synthetic but representative gas-lifted oil eld modeled in a standard simulator.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

The increasing demand for petroleum and the maturing ofexisting oil elds have compelled oil operators to invest in newtechnologies to optimize their production processes and cut ope-rating costs. Further, the high costs of drilling and operating res-ervoirs put pressure on the operators to yield early returns on theinvestments, particularly so in the reservoirs of the Pre-Salt layerlocated off the coast of Brazil. To this end, the oil companies seekto optimize daily operating plans which consist of gas-lift injectionrates, production choke openings, well-manifold routings, andpressures, among others. Such initiatives are aligned with theconcept of Smart Fields (Yeten et al., 2004; Camponogara et al.,2010) which aim to drive production and economic gains byeffectively integrating subsea equipment, control and informationsystems, and optimization software.

Because Smart Fields is an evolving technology, eld engi-neers still rely on sensitivity analysis using simulation softwareand heuristics to decide upon the daily operational plans andrespond to unanticipated events, such as compressor failure andpipeline clogging. However, this strategy can be rather time-consuming and does not necessarily ensure a mode of operationthat maximizes the daily production.

An alternative that is gaining acceptance in the industry ismodel-based optimization, which can be viewed as the integra-tion of mathematical models with algorithms into effectiveoptimization tools. Such models should be routinely updatedwith eld data to reect the prevailing system conditions. Forsteady surface conditions and satellite wells, models and algo-rithms have appeared in the technical literature (Buitrago et al.,1996; Fang and Lo, 1996; Alarcn et al., 2002; Camponogara andde Conto, 2009; Misener et al., 2009; Codas and Camponogara,2012). On the other hand, more complex models have beenproposed to account for varying operating conditions, which aretypical of production systems with subsea completion (Litvakand Darlow, 1995; Kosmidis et al., 2004, 2005; Gunnerud andFoss, 2010; Codas et al., 2012; Silva et al., 2012). For instance, theproduction of wells can be gathered in a subsea manifold beforeowing to an offshore oil platformtherefore, the well-headpressure and pressure drop in the well jumper will depend onthe manifold to which the well is connected.

In modern offshore oil elds, wells are often equipped withrouting valves to direct their production to multiple manifoldheaders, a strategy that is routinely adopted in practice either toprovide resilience to equipment failure or to adjust the well-manifold routings to improve production. Despite ow splitt-ing being a common practice in industrial settings, to the best ofour knowledge it is not accounted for by existing mathematicalmodels in the optimization literature, whose works usually imp-ose a single routing from wells to manifolds. Incidentally, ow

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Journal of Petroleum Science and Engineering

http://dx.doi.org/10.1016/j.petrol.2015.02.0180920-4105/& 2015 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: 55 48 3721 7688; fax: 55 48 3721 9934.E-mail address: [email protected] (E. Camponogara).

Journal of Petroleum Science and Engineering 128 (2015) 8697

splitting induced by routing well production to multiple mani-folds is a common practice in the Urucu eld, a reservoir locatedin the heart of the Amazon, which is not addressed by the exi-sting models (Codas et al., 2012). To this end, this paper adv-ances previous works by proposing a model that decides uponthe splitting of ows in pipelines.

The paper is organized as follows. Section 2 proposes a mathe-matical model for ow splitting according to the observed behaviorof a commercial multiphase-ow simulator. Section 3 approximatesthis model with piecewise-linear functions which are then validatedagainst the simulator. Section 4 discusses the modeling of a synthetic,but representative eld, and proposes a methodology to obtain suf-ciently accurate Piecewise-Linear (PWL) models for well-productionand pressure drops. Section 5 evaluates the performance of the PWLformulation and analyzes the impact of ow splitting. The concludingremarks are presented in Section 6.

2. Flow splitting modeling and validation for productionoptimization

A typical offshore production system is composed of subsea wells,manifolds gathering production from wells, and surface facilities. Pro-duction wells are equipped with choke valves that control well-headpressure and production, which can benet from gas-lift to increasethe ow rate. After being gathered by a manifold, the well productionis directed to a surface separator which splits the production streaminto three-phase ows, namely, oil which is transferred by shuttletankers to an onshore terminal, gas which is compressed and exportedin subsea pipelines, and water which is processed before discharge.

Fig. 1 illustrates a subsea production system consisting of asingle well and three manifolds gathering production. The pres-sures and ow rates depicted in the gure correspond to the well-head pressure (pnwh), the pressure downstream the choke (p

nds), the

manifold pressure (pm), the well production (qn), the ow rates inthe jumpers (qn;m), and the lift-gas rate (qninj).

In such elds, operators usually decide upon the routing ofproduction from wells to manifolds, which are implemented byopening or closing the routing valves. To the best of our knowl-edge, previous works found in the technical literature enforce thepolicy of routing wells to a single manifold (Gunnerud and Foss,2010; Codas et al., 2012), despite multiple routing being routinelyimplemented in real-world oil elds.

In what follows, a mathematical nonlinear model for ow spli-tting is developed in the context of a single well and multiple man-ifolds. This model can be used to represent ow splitting of wells in

complex production networks encompassing several wells and mul-tiple manifolds.

2.1. Nonlinear model

Consider a particular gas-lifted well n of a set N f1;;Ng.Suppose that well n can send its production stream to a subsetMn ofthe manifolds, with MnDM f1;;Mg. The oil, gas, and waterproduced by well n can be characterized by the following equations:

qnoil bqnliqpnwh; qninj 1WCUTnqngas bqnliqpnwh; qninj GLRnqninjqnwater bqnliqpnwh; qninj WCUTn

8>>>>>: 1with bqliq being the liquid production as a function of the well-headpressure pnwh and the rate of the lift-gas injection q

ninj. For the purpose

of steady-state production optimization, the gasliquid ratio (GLR)and the water cut (WCUT) are assumed known and constant over thehorizon of production planning.

The difference between the well-head pressure and the pressuredownstream the choke, pnds, corresponds to the pressure loss due tofriction which is related to a particular choke opening. Flows in thejumper connecting well n to manifold m are given by functions asfollows:

qnsp bqnspqn; pnds;pnman; 2aqn

XmAMn

qn;m; 2b

qm X

nANmqn;m; 2c

pnwhZpnds; 2d

pnds dpn;mjp qn;mliq ;GORn;mpm; mAMn 2eGORn;m 1WCUTn GLRn

qninjqnoilqnwater

!; 8mAMn 2f

where

1. qn qnoil; qngas; qnwater is a vector with the three-phase owproduced by well n,

2. qnsp qn;moil ; qn;mgas ; qn;mwater : mAMn

is a vector with the rate of alluid phases owing in the jumpers,

3. pnman pm : mAMn is a vector with the pressure of the man-ifolds receiving the production stream from well n.

The three-phase ows in the jumpers are given by the functionbqnspqn, pnds, pnman which depends on the total ow rate qn of well n,the pressure downstream the choke (pnds), and the pressures at themanifolds to which the well is connected (pnman). A multiphase-owsimulator iteratively calculates the ow for each jumper based on thepressure differences downstream the choke to the manifold, keepingthe same gasliquid ratio of the well in accordance withEq. (2f). Notice that the left-hand and the right-hand side of Eq.(2f) dene a factor that, whenmultiplied by a ow of liquid, producesthe respective ow of gas. This factor is the gasliquid ratio (GLR) forthe jumper on the left-hand side and for the well on the right-handside. The pressure drops in the jumpers are calculated by the

function dpn;mjp , being induced by the ows.Typically, the functions bqnsp and dpn;mjp are not known explici-

tly but rather implemented by simulation software, which iterativ-ely converges to a solution of the system of equations (2) that

Manifolds

Jumpers

Well

Choke

Routing Valves

Tubing

Compressor

Fig. 1. Flow splitting illustration.

T. Lima Silva et al. / Journal of Petroleum Science and Engineering 128 (2015) 8697 87

meets Physics laws. More specically, Eq. (2f) ensures that the gasand liquid ow rates of the jumpers are in the same proportion ofthe ows produced by the well, a behavior observed in commer-cial simulators.

The splitting model given in Eq. (2) is based on three principles.The rst principle is the mass conservation of the ow leaving awell which is split in the ows entering the manifolds. The secondprinciple is that pressure differences between well-head and man-ifolds are established by pressure-drop functions that depend onthe ows and the physical properties of the pipelines. The thirdprinciple is the assumption that the gasliquid ratio of the well-production ow is the same of the streams reaching the manifolds.

2.2. Piecewise-linear approximation

Despite being routinely used by reservoir engineers to predictproduction, the relations implemented by simulation software areeither not explicitly known or too complex to be effectively used inmathematical optimization. Alternatively, the relations implemen-ted by simulators can be conveniently modeled with multidimen-sional piecewise-linear functions, such as the well productionsurface and its corresponding piecewise-linear model illustratedin Fig. 2(a) and (b). The function in a particular polytope, namelyP1, is described through the convex combination of the vertices ofthe polytope and their corresponding function values:

xP1 X4i 1

i vi;X4i 1

i 1

f xP1 X4i 1

i f vi; iZ0; i 1;4

where i is the weighting variable associated with vertex vi. Noticethat a piecewise-linear function can approximate a nonlinear fun-ction to a desired degree of accuracy provided a sufcient numberof sample points.

The advantage of piecewise-linear models with respect to others isthat the former are directly obtained from the sample data, dispen-sing with the synthesis of proxy models, a task that can itself berather complex.

Optimization problems involving these functions can be mod-eled as Mixed-Integer Linear Programs (MILP) and solved with spe-cialized algorithms or general-purpose solvers. Usually, the latterapproach takes advantage over the rst since it uses the advan-ced technology available for solving MILPs (Vielma et al., 2010).

A comprehensive study of available MILP models to representmultidimensional piecewise-linear functions in the context of oilproduction optimization is found in Silva and Camponogara (2014).

In what follows, the main idea of the paper is presented, whichconsists of approximating implicit relations used by multiphase-ow simulators to predict splitting of ows as piecewise-linearmodels. This methodology is then used to optimize the productionof a representative offshore gas-lifted oil eld.

2.2.1. MILP approximation for splittingThe mathematical modeling of ow splitting developed in

Section 2.1 is interesting for understanding the process, but notappropriate for optimization purposes since it relies on implicitfunctions implemented by simulation software. Herein we approx-imate such functions using the multidimensional piecewise-linearmodel based on specially ordered sets of variables of type 2 (Beale,1980; Tomlin, 1988) denoted by SOS2, resulting in an MILP formula-tion. This model was chosen for its simplicity and efciency forhypercube domains (Silva and Camponogara, 2014). A short note onthe SOS2 model appears in Appendix A.

The well production curve bqnliqqninj; pnwh is approximated with apiecewise-linear (PWL) function ~qnliq ~qnoil ~qnwater. Gas and waterproduction are obtained from the gasliquid ratio and water-cutrelations. First, Eq. (1) is recast as

~qnoil P

qi ;pkAKnnqi ;pk bqnliqqi; pk 1WCUTn;

~qngas P

qi ;pkAKnnqi ;pk bqnliqqi; pk GLRnqninj;

~qnwater P

qi ;pkAKnnqi ;pk bqnliqqi; pk WCUTn;

8>>>>>>>>>>>>>:3

where Kni is the set of breakpoints of lift-gas injection rates, Knp isthe set of breakpoints of well-head pressure of well n, and KnKni Knp. Further, nqi ;pk is the weighting variable associated withthe breakpoint qi; pkAKn.

The gas-injection rate and the well-head pressure of well n aredened with the same weighting variables nqi ;pk used to approx-imate the production function:

qninj X

qi ;pkAKnnqi ;pk qi; 4a

pnwh X

qi ;pkAKnnqi ;pk pk: 4b

1500

2000

2500

3000

3500

200250

300350

150

200

250

300

350

400

450

500

Gasl

ift (sm

3/d)

Wellhead pressure (psia)

Liqu

id F

low

Rat

e (m

3/d)

1500

2000

2500

3000

3500

200250

300350

150

200

250

300

350

400

450

500

Wellhead pressure (psia)Ga

slift (

sm3/d

)

Liqu

id F

low

Rat

e (m

3/d)

Fig. 2. Illustration of a well production function and a piecewise-linear approximation. (a) Well production surface and (b) piecewise-linear illustration.

T. Lima Silva et al. / Journal of Petroleum Science and Engineering 128 (2015) 869788

Additional constraints are added to implement the piecewise-linear model SOS2 approximating the well production function(bqnliq):1

Xqi ;pkAKn

nqi ;pk ; 5a

0rnqi ;pk ; 8qi; pkAKn; 5b

qi X

pkAKnpnqi ;pk ; 8qiAK

ni ; 5c

pk X

qiAKninqi ;pk ; 8pkAK

np; 5d

qi qiAKi ; pk pkAKp are SOS2; 5e

where qi and pk are auxiliary variables which are used to imp-lement SOS2 constraints.

The pressure drop dpn;mjp qn;mliq ;GORn;m in the jumper connect-ing a well n to manifoldm is approximated with a piecewise-linearmodel:

~pn;mjp X

ql ;gorARn;mn;mql ;gor dpn;mjp ql; gor 6

where Rn;mRn;ml Rn;mg is the set of breakpoints for liquid owrates (Rn;ml ) and gasoil ratio values (Rn;mg ) in the jumper, withn;mql ;gor being the weighting variable associated with each break-point ql; gor in the set Rn;m.

Resulting ows split from a particular well n are calculatedwith the weighting variables n;mql ;gor associated with the pressure-drop approximation ~pn;mjp :

qn;moil P

ql ;gorARn;mn;mql ;gor ql 1WCUT

n;

qn;mgas P

ql ;gorARn;mn;mql ;gor ql 1WCUT

n gor;

qn;mwater P

ql ;gorARn;mn;mql ;gor ql WCUT

n:

8>>>>>>>>>>>>>:7

Notice that the ows in the jumpers are now explicitly calculatedbased on the pressure gradients established by the piecewise-linear approximation of dpn;mjp . Then, gasliquid proportions ofsplit ows are kept the same of the well in accordance with anapproximation of Eq. (2f) as follows:Xql ;gorARn;m

n;mql ;gor gor 1WCUTn GLRn

Xqi ;pkAKn

nqi ;pk qibqnliqqi; pk:

8Further constraints are added to implement the SOS2 pie-

cewise-linear model that approximates the pressure drop in thejumpers:

1X

ql ;gorARn;mn;mql ;gor ; 9a

n;mql ;gorZ0; 8ql; gorARn;m; 9b

n;mql X

gorARn;mgn;mql ;gor ; 8qlAR

n;ml ; 9c

n;mgor X

qlARn;ml

n;mql ;gor ; 8gorARn;mg ; 9d

n;mql qlARn;ml ; n;mgor gorARn;mg are SOS2: 9e

The auxiliary variables n;mql and n;mgor are used to implement the

piecewise-linear strategy with SOS2 variables.

The proposed model of ow splitting can also be developedwith other piecewise-linear formulations, such as the convexcombination models (DCC, DLog, CC, and Log), the incrementalmodel, and the multiple choice model (Silva and Camponogara,2014). Also, the splitting of the production stream of wells thatoperate without articial lifting, such as in the gas reservoir of theMexilho Field located off the coast of Brazil, can be easily modeledby removing the lift-gas contribution from Eqs. (3) and (8).

2.3. Simulation-based validation

The validation of the mathematical model of ow splitting iscarried out by contrasting its behavior against the behavior obs-erved in a steady-state multiphase ow simulator. For this pur-pose, a small synthetic eld is developed and used as a testbed forthe experiments.

2.3.1. Synthetic eld modelingFig. 3 illustrates a gas-lifted well instantiated in the multiphase

ow simulator Pipesim from Schlumbergers. The ow rate of thewell can be increased by injecting pressurized gas at the bottom ofthe production tubing and by choking production at the well-head,which changes the back-pressure to the well-head.

The well has the following attributes: Liquid Productivity Index(PI)80 STB/d/psi, gasliquid ratio (GLR)160 sm3/sm3, Water-Cut(WCUT)0.01 sm3/sm3, and reservoir static pressure of 2100 psia. Itis assumed that the reservoir characteristics do not vary frequently,thus the gas-liquid ratio, water-cut, and productivity index of thewell are constants for the horizon of production optimization.

The well is connected to subsea manifolds that represent thesubmarine equipment, which are typically found in offshore oilelds.Fig. 4 illustrates the subsea equipment of a simple offshore eld,which consists of a single well owing production to three subseamanifolds through jumper pipelines with 4 in of inner diameter,0.25 in of wall thickness, and 0.001 in of roughness. The pipelinesB15, B25, and B2 have different lengths, namely 1 km, 1.25 km, and1.5 km, respectively.

2.3.2. Simulation analysisIn this section, a simulation analysis is performed with the goal of

evaluating whether the MILP model approximates satisfactorily theow splitting observed in the commercial multiphase ow simulator.

The methodology adopted to assess the degree of accuracy of theMILP model is illustrated in Fig. 5. The methodology consists of thefollowing steps:

Step 1: Boundary conditions such as pressures in the manifoldsand lift-gas rates are given as inputs for the simulator, which iter-atively calculates the pressures in the well-head and downstream thechoke, along with the resulting ow rates in the pipelines.

Step 2: Flow rates and pressures calculated by the simulator(qSIM, pSIM) are given as references for the splitting model.A feasible solution is calculated by the splitting model for which

Fig. 3. Gas-lifted well in pipesim.


the innity norm pSPLpSIM1 is bounded by the tolerance . Ifthe difference between the ow rates and pressures estimated bythe splitting model and the simulator is smaller than the tolerance,a reasonable solution was achieved and the procedure halts,otherwise it continues from step 3.

Step 3: The well-production and pressure-drop approximationsare rened by sampling well ow rates and pressure drops fromthe simulator for a higher number of breakpoints. The new PWLfunctions are given as inputs for the splitting model and the proc-edure continues on from step 2.

The procedure was initialized with PWL approximations contain-ing 5 breakpoints per axis and having qrefinj , prefman 5200; 150;155;1600 as the boundary conditions. A solution was reached after3 iterations, with the resulting approximation for the well-pro-duction function (bqnoil) having a total of 289 breakpoints, being 17for injection ow rate and 17 for well-head pressure. The resulting

approximation for pressure-drop functions (dpn;mjp ) also contains 289breakpoints, namely 17 per axis.

The results of the analysis appear in Table 1, which presents owrates in cubic meters per day (sm3/d) and pressures in psia. Noticethat the maximum error between the simulator outputs and thesplitting model predictions is less than 2.5%, which was the chosentolerance for convergence of the procedure.

Despite the errors, the well ow rates predicted by the optimizerare similar to the ows calculated by the simulator (error r0:40%)and the pressure differences are acceptable (error r1:49%). Theow rates produced by the simulator and the approximation modelare similar for manifolds 1 and 3 (error r0:65%). The prediction

errors are higher for manifold 2, but still acceptable. The highesterrors were observed in the predictions of pressure, which never-theless did not exceed the upper bound on errors (2.5%).

Despite the model discrepancies, the simulation analysis showedthat the proposed model can satisfactorily approximate the ow-splitting phenomenon observed in the simulator.

3. Application to production optimization

A mathematical methodology to approximate ow splitting insubsea equipment was developed and validated against a commer-cial multiphase-ow simulator. This methodology is now incorpo-rated into a mixed-integer programming model to optimize a repr-esentative offshore production system with multiple routing deci-sions and lift-gas distribution.

A typical offshore oileld consists of wells that drain uids from areservoir to subsea manifolds which gather production to the com-pression and separation system of a Floating Production Storage andOfoading (FPSO) platform, as illustrated in Fig. 6. The ow paths fromwells to manifolds are determined by routing valves. When the ow

Fig. 4. Illustration of the pipesim model of a gas-lifted oil well connected to threemanifolds.

Boundary Conditions

Lift-gas Rate ( )Pressures of Manifolds ( )

SimulatorIteratively Calculates:

PressuresFlows in pipelines

1

Splitting ModelFind a feasible solution

for a small tolerance2

Such that:

Sampling Program

Refine PWL Models:

Production FunctionPressure Drop Functions

3

Error (%) > tolerance

Solution

Error tolerance

PWL fu

nctions

Fig. 5. Flow-splitting simulation analysis workow.

Table 1Simulation analysis of ow splitting.

Variable Splitting Model Simulator Error (%)

Well

qngas 96972.70 97365.27 0.40qnoil 563.45 565.32 0.33

qnwater 5.69 5.71 0.35pnwh 192.83 190.00 1.05pnds 192.83 190.00 1.05qninj 5910.56 6000.00 1.49

GLRn 170.39 170.51 0.07

Manifold 1

qmgas 39644.60 39710.47 0.17qmoil 230.35 230.57 0.00

qmwater 2.33 2.33 0.00pm 147.86 150 1.43

GLRn;m 170.39 170.51 0.07

Manifold 2

qmgas 32313.50 31613.81 2.21qmoil 187.48 183.56 2.14

qmwater 1.89 1.85 0.00pm 153.15 155.00 2.16

GLRn;m 170.39 170.51 0.07

Manifold 3

qmgas 25893.60 26040.99 0.57qmoil 150.45 151.20 0.50

qmwater 1.52 1.53 0.65pm 157.62 160.00 1.49

GLRm 170.39 170.51 0.07


arrives at the platform, the separation system removes the gas fromthe mixture which is then compressed and exported to an onshoreunit or used for gas-lift. The water is treated before discharge, whilethe oil is stored and then transferred to the coast in shuttle tankers.

The problem of optimizing the production of an offshore eldsubject to gas-lift distribution, pressure constraints and multiplerouting decisions is considerably hard to solve due to the non-linear nature of the production and pressure-drop functions, alongwith the discrete variables concerning well activation and routing.To this end, the problem is formulated as an MILP program usingpiecewise-linear models to approximate the nonlinear functions.

Despite being a common practice in real-world oil elds, pre-vious mathematical models avoid dealing with scenarios in whichow splitting takes place due to the complexity involved. The pro-posed MILP model is able to automatically decide upon the spli-tting of ows in the subsea pipelines, representing the division ofuids according to commercial simulation software.

3.1. Mixed-integer linear programming model

In this section an MILP formulation is developed to optimize thedaily production of an offshore production system subject to gas-liftdistribution and multiple-routing decisions. The splitting model isincorporated into this formulation, with small changes to supportrouting decisions and variations on the manifold pressure. The listof sets, parameters, variables, and functions of the MILP formulationis available in Tables C1, C2, C3, and C4 of Appendix C, respectively.

The goal of the optimization problem is expressed in theobjective function f as the maximization of the total oil producedin the manifolds:

max f X

mAMqmoil: 10

Although more complex objectives could be readily used by means ofpiecewise-linear approximation, oil production remains widely adoptedarguably for being more easily measured in real-world settings.

The production of the wells bqnliq is given by a piecewise-linearfunction which approximates the ow rates observed in the simulator,obtained by sampling the production functions for a sufciently wide

range of lift-gas rates and well-head pressures. The well productionapproximation equations are omitted here for simplicity, since theyare similar to the PWL approximation developed in previous section.For more details, refer to Eqs. (B.1a)(B.1k) in Appendix B.

Binary variables (yn) are introduced to express the possibility ofshutting in wells, a procedure that may be required for operationalreasons or to improve overall system production.

When a well is active (yn1) its production is bounded by ope-rational limits:

ynqn;Lrqnrqn;Uyn; 8nAN : 11The well-manifold ow paths are determined by the binary

variables zn;m. Notice that the algorithm will decide upon single ormultiple routing. Routing fromwell n to manifoldm (zn;m) can only beactive when the well is producing (yn1). Otherwise (yn0), therouting from this well is disabled (zn;m 0; 8mAMn). Further, inorder to account for the manifold pressure variation in the model, thepressure drops become also dependent of the pressure downstreamthe choke (pnds). For being similar to the PWL approximation developedin the splitting model, the pressure-drop approximation equations areomitted here, but fully dened in Eqs. (B.2a)(B.2o) in Appendix B.

The mass balance of ows produced by the wells and split tothe manifolds are imposed by the following vector constraints:

~qn X

mAMnqn;m; 8nAN ; 12a

qm X

nANmqn;m; 8mAM: 12b

Each manifold can handle certain rates of oil, gas, and water whichare honored by constraints bounding all of the phase ows:

qmrqm;max; 8mAM: 13The platform limits on compression, liquid handling, water tre-

atment, and gas-lift are imposed as bounds on the total productionof gas, liquid, and water:

For all mAM:XmAM

qmgasrqmaxg ; 14a

Fig. 6. Illustrative offshore gas-lifted oileld.


XmAM

qmoilqmwaterrqmaxl ; 14b

XmAM

qmwaterrqmaxw ; 14c

XnAN

qninjrqmaxinj : 14d

Pressure constraints on subsea equipment are then established:

For all nAN :pnwhZp

ndspn;maxwh 1yn; 15a

pndsrpm ~pn;mjp pn;maxds 1zn;m; 15b

pndsZpm ~pn;mjp pn;maxds 1zn;m; 15c

pndsrpn;maxhipps : 15d

A High-Integrity Pressure Protection System (HIPPS) ensures thesafety of the production system, establishing a bound (pn;maxhipps ) on

the pressure downstream the choke. Notice that the differencebetween the well-head pressure (pnwh) and the pressure down-stream the choke (pnds) gives the pressure loss in the choke.

Pressure drops in the owlines, which rise the production frommanifolds to platforms, are calculated by a PWL approximation ofthe pressure-drop function dpmqmliq;GORm;WCUTm:For all mAM:~pm

Xqliq ;gor;wcutAQm

mqliq ;gor;wcut dpmqliq; gor;wcut; 16aqmoil

Xqliq ;gor;wcutAQm

mqliq ;gor;wcut qliq 1wcut; 16b

qgas X

qliq ;gor;wcutAQmmqliq ;gor;wcut qliq gor 1wcut; 16c

qmwater X

qliq ;gor;wcutAQmmqliq ;gor;wcut qliq wcut; 16d

ymman X

qliq ;gor;wcutAQmqliq ;gor;wcut ; 16e

qliq ;gor;wcutZ0; 8qliq; gor;wcutAQm; 16f

zn;mrymman; 8nANm; 16g

mqliq X

gorAQmg

XwcutAQmw

qliq ;gor;wcut ; 16h

mgor X

qliqAQml

XwcutAQmw

qliq ;gor;wcut ; 16i

mwcut X

qliqAQml

XgorAQmg

qliq ;gor;wcut ; 16j

mqliq qliqAQl ; mgorgorAQg ;

mwcutwcutAQw are SOS2: 16k

The binary variable ymman denotes the activation of manifold m.Notice that the pressure drop in the owlines depends on the liq-uid ow rate and the proportions of gas and water of the mixture.

The manifold pressure (pm) must be equal to the separatornominal pressure (pm;S) to which it is connected plus the pressuredrop in the ow line ( ~pm):

pm pm;S ~pm; 8mAM: 17

Putting all together, the problem of allocating pressurized gasand deciding upon the routing of wells to separation units is exp-ressed in a compact form as

P :

max f PmAM

qmoil

s:t: : Constraints B:1aB:1k;Constraints 1114;Constraints B:2aB:2o;Constraints 1517:

8>>>>>>>>>>>>>:

4. Model synthesis and simulation analysis

This section evaluates the developed framework for ow spli-tting in a representative offshore oil eld which operates with gas-lift, allows splitting of ows, and is further constrained by physi-cal and operational constraints. A methodology is proposed for thesynthesis of piecewise-linear models that satisfactorily approxi-mate the nonlinear process functions.

4.1. The gas-lifted oil production system

The synthetic production systemwas inspired in Kosmidis et al.(2005) and Silva and Camponogara (2014) and modeled in a multi-phase-ow simulator, namely Schlumbergers Pipesim. This sys-tem will serve as a testbed for model synthesis and computationalanalysis. Fig. 7 illustrates the production infrastructure of thisoil eld.

The wells are topologically divided into three groups:

1. wells 12 are 1 km away from manifold 1 and 1.5 km frommanifold 2;

2. wells 35 are 1 km away from both manifolds;3. wells 67 are 1.5 km away from manifold 1 and 1 km from

manifold 2.

The pipelines called jumpers are those connecting wells tomanifolds, with 4 in of inner diameter (ID), 0.25 in of wall thick-ness (W), and 0.001 in of roughness (R). The jumpers with 1 and1.5 km of length are denoted by J1 and J2, respectively.

The pipelines called owlines are the ones sending the productionof the manifolds to the platforms. All pipelines have 5.5 in of innerdiameter, 0.5 in of wall thickness (W), and 0.001 in of roughness. Theowlines F1 and F2 have 2.5 km and 2 km of length, respectively. Eachmanifold has a dedicated platform for sending production. Manifold1 is connected to the platform by pipeline F1, while manifold 2 is con-nected to its dedicated platform by pipeline F2.

Some assumptions were made on the eld simulation model: thereservoir has a constant pressure; GLR andWCUT of wells do not varyduring the optimization process; after well ow is split, resultingows have the same GLR of the ow before splitting. The liquid owrate of wells behaves according to the equation ql piprpwf where pwf is the bottom hole pressure, pi is the well productionindex, and pr is the reservoir static pressure. Well parameters such asGLR, WCUT, pr, and pi are shown in Table 2, with the units beingsm3=sm3, %, psi, and STB/d/psi, respectively.

The absolute pressure in the manifolds ranges from 275 to575 psi depending on the operational conditions, while the nom-inal pressure at the separators located in platforms 1 and 2 are 125and 150 psi, respectively.

4.2. Model synthesis

This section presents a procedure to synthesize mathematicalmodels for well-production and pressure-drop functions adjusted


for the synthetic production system. The procedure consists ofanalyzing the errors of each approximation by comparing the sam-pled function values and the values calculated by the math-ematical model.

4.2.1. Grid-tting procedureIn Silva and Camponogara (2014), it is shown that the errors in

the optimizer variables can be reduced by introducing more break-points in the piecewise-linear approximations, with the disadvantageof increasing computational time and complexity of the resultingproblem. This process was generalized by Aguiar et al. (2014) withthe development of a simple off-line procedure to reduce the disc-repancy between optimizer predictions and simulator estimates.

Although these methods can improve the quality of approxima-tion, they usually increase excessively the model complexity, becauseunnecessary breakpoints are added to the domain in each step. Inthis work, we propose a heuristic procedure based on an algorithmproposed by Codas et al. (2012) to obtain suitable approximations foroptimization purposes.

The procedure is outlined in the following steps and illustratedwith the example depicted in Fig. 8:

1. Fig. 8(a) presents a two-dimensional domain with 4 polytopes,namely P fP1, P2, P3, P4g for a given piecewise-linear approx-imation ~f x of a nonlinear function f x. This approximation isgiven as input for the renement procedure.

2. Fig. 8(b) illustrates the renement step performed by the grid-tting procedure. The functions values f x of central pointsx1; y1, x2; y1, x1; y2, and x2; y2 are sampled from thesimulator. The function values of the piecewise-linear approx-imation ~f x are then calculated for the same points by theconvex combination of corner vertices. The relative error is

obtained by measuring the deviation of the approximation ineach polytope:

Pi;j j ~f xi; yj f xi; yjj

f xi; yj; i; jA 1;2f g

If the maximum error maxfP1;1; P1;2; P2;1; P2;2g is lower than thetolerance, then no breakpoints are introduced and the PWLapproximation is considered satisfactory for optimization pur-poses. Otherwise, new breakpoints are added in the polytopeswhere the estimated error is higher than the tolerance.

3. Fig. 8(c) shows that the resulting domain after the renementstep is performed for an approximation in which only polytopeP3 presents an estimated error higher than the tolerance. Thispolytope was subdivided into 4 polytopes fP3a, P3b, P3c, P3dg,while polytopes P1 and P4 were subdivided into fP1a, P1bg andfP4a, P4bg, respectively. The subdivision of polytopes P1 and P4 isdue to the introduction of the new breakpoints x1 and y2 in thePWL approximation, which is composed of the Cartesian pro-duct of all breakpoints from both axis. The subdivision of onlypolytope P3 would be possible with other models for PWL suchas CC and DCC (Vielma et al., 2010), instead of SOS2 constraints.

This procedure is a simple way for estimating the approximationerror of a general PWL function with hypercube domains. It caneasily be extended to higher order and different domains.

4.2.2. Model analysisThe grid-tting procedure is now used to obtain suitable app-

roximations for the synthetic production system. The well-pro-duction function bqnoil was approximated with the procedure illu-strated above, while the pressure drops dpn;mjp and dpm wereapproximated with an extension to three-dimensional domains.

The initial approximation for the well-production ~qnoil has 5breakpoints in both domain axes: lift-gas rate and well-head pre-ssure. Table 3 shows the errors of the nal approximations acc-ording to the tting procedure.

jKnqi j and jKnpkj are the number of breakpoints for lift-gas rate

and well-head pressure, respectively. The approximations of well-production functions have a maximum error of 1.86%, mean errorand standard deviation are less than 0.50%.

The starting domain of the jumper pressure-drop functionsdpn;mjp contains 125 points with 5 breakpoints in each axis (i.e., ql,gor, and pds). Table 4 shows the resulting errors of the approxima-tions produced by the tting procedure. The number of break-points for liquid ow rate, gasoil ratio, and pressures down-stream the choke are represented by the cardinality of the setsjRn;ml j , jRn;mg j , and jRn;mp j , respectively. The nal approximationshave a maximum error less than 1.88%, and a mean error under1.30%, which are smaller than the tolerance of 2.00%. Notice thatthe approximations for the pressure drops in the jumpers wereobtained with fewer iterations than the approximations for thewell-production functions.

Finally, Table 5 shows the approximation errors of the pre-ssure-dropdpm functions in the owlines. jQml j , jQmg j , and jQmw jare the number of breakpoints for liquid ow rate, gasoil ratio,and water cut values, respectively. The maximum errors are small(o0:3%), while the mean errors and standard deviations are neg-ligible (o0:03%). For these approximations only 2 iterations wereneeded to reach an approximation with small errors.

The grid-tting procedure was able to nd approximationswith errors within the tolerance for the well-production and pres-sure-drop functions, after a small number of iterations. This resultshows that the nal approximations are representing the well-production and pressure-drop functions satisfactorily.

1 2

1 2Separators

Manifolds

Wells

Compressor

Gas-lift Manifold

1,2 3 5 6,7

Flowlines

Jumpers

Fig. 7. Production system network with gas-lifted wells.

Table 2Well parameters.

Well GLR WCUT pr pi

1 70 1.00 2400 222 52 2.00 2650 253 62 1.00 2550 294 60 1.50 2500 305 65 2.00 2450 276 70 1.50 2600 317 55 1.50 2350 23


5. Computational analysis

This section presents a computational analysis of the performanceof the MILP formulation developed for production optimizationwhichconsiders the inuence of ow splitting. The production system of theprevious section will serve as the testbed for the experiments.

For the purpose of comparison of the MILP formulations withand without ow splitting, single well-manifold routing isenforced in the MILP formulation (P) by introducing the followingconstraint on the binary variables zn;m:XmAMn

zn;mr1; 8nAN : 18

Both formulations were expressed in AMPL (Fourer et al., 2002)and solved with the MILP solver CPLEX 12.6 in a Linux workstation,using an Intel Xeon E5-2665 processor at 2.40 GHz and 40 GB ofRAM. After the pre-solve step, the resulting formulations have thefollowing properties:

1. Automatic routing: 10 060 variables (14 binary), 794 linearconstraints, and 62 SOS2 constraints.

2. Single routing: 10 060 variables (14 binary), and 801 linearconstraints, and 62 SOS2 constraints.

Notice that with automatic routing, formulation P will decide foreach well on the splitting of ows (whether its production will besent to one or more manifolds), whereas single routing will forcethe production to ow to a single manifold.

The analysis evaluates the performance of both automatic andsingle routing models for three availabilities of lift-gas:

1. High: The available gas rate is sufcient to inject the maximumrate allowed in all wells simultaneously (24 500 sm3/d).

2. Medium: The average gas rate availability (15 500 sm3/d).3. Low: Smallest gas rate availability which enables the opening of

all wells (14 500 sm3/d).

All experiments ran within a time limit of 30 min. Table 6 shows theresults obtained by the automatic and single routing formulations.

With the increase of the lift-gas availability, the optimal oilproduction increased slightly for both formulations. All solutionsfound by the automatic routing model induced splitting of owsand yielded higher oil production rates (1.502.00%) in comparisonto single routing. The single routing model was solved more easily,requiring a reduced number of branch-and-bound nodes to reachthe optimal solution for the scenarios with medium and low lift-gasavailability. For these scenarios, the solver did not nd the optimumfor the automatic routing model within the time limit of 30 min.Nevertheless the best solutions found yielded higher oil productionrates than the optimal solutions found with the single routingmodel. On the other hand, the automatic routing model was solvedmore expeditiously for the scenario with high lift-gas availability.

6. Summary

This work proposed a mathematical model to represent thesplitting of ows which is of particular interest in subsea opera-tions. The splitting model was approximated by an MILP modelbased on multidimensional piecewise-linear functions and vali-dated by contrasting its predictions against what is observed insimulation software.

An MILP formulation was developed for the problem of max-imizing the production of a representative offshore oileld subjectto lift-gas distribution, pressure constraints, and multiple routingdecisions. Further, a heuristic procedure was designed to obtainwell-production and pressure-drop approximations with mean

Fig. 8. Procedure to rene piecewise-linear functions. (a) Given domain, (b) renement step, and (c) resulting domain.

Table 3Well-production approximations.

Well jKnqi j jKnpkj Max Error Mean Error Std. Deviation Iterations

(%) (%) (%)

1 13 15 1.34 0.28 0.29 52 9 11 1.67 0.43 0.38 43 9 9 0.94 0.25 0.25 44 13 15 1.12 0.33 0.33 55 9 9 1.86 0.45 0.45 46 5 5 0.73 0.26 0.26 27 5 5 1.60 0.44 0.44 2

Table 4Jumper pressure drop approximations.

Well Man. jRn;ml j jRn;mg j jRn;mp j Max Err. Mean Err. Std. It.(%) (%) (%)

1 1 9 9 9 1.49 086 0.33 32 9 9 9 1.54 0.90 0.34 3

2 1 9 9 9 1.61 0.95 0.37 32 9 9 9 1.63 0.96 0.36 3

3 1,2 5 5 5 1.15 0.74 0.25 24 1,2 9 9 9 1.60 0.98 0.37 35 1,2 5 5 5 1.41 0.87 0.32 26 1 9 9 9 1.63 0.96 0.35 3

2 9 9 9 1.48 0.87 0.31 37 1 9 9 9 1.88 1.30 0.43 3

2 9 9 9 1.48 0.87 0.31 3

Table 5Pressure drop approximations in the owlines.

Manifold jQml j jQmg j jQmp j Max Err. Mean Err. Stds. It.(%) (%) (%)

F1 5 5 5 0.29 0.007 0.03 2F2 5 5 5 0.09 0.004 0.02 2


and maximum errors within a given tolerance. The model analysisshowed that the approximation errors become smaller than thetolerance after a few iterations of the procedure. This result indicatesthat the ow-splitting model can accurately reproduce the phenom-ena observed in multiphase ow simulation. Finally, the computa-tional analysis showed that the standard single-routing model issolved faster than the automatic-routing model, but the latter rea-ched solutions with higher overall oil production rates.

Future research include the application of the automatic rout-ing model and approximation tools to more complex productionsystems and the use of other approximation models for well-production and pressure drop.

Acknowledgments

This research was supported in part by Petrleo Brasileiro S.A.(Petrobras) and Conselho Nacional de Pesquisa e DesenvolvimentoTecnolgico (CNPq).

Appendix A. SOS2 model

In a pioneering work, Beale and Tomlin (1970) proposed amodel torepresent piecewise-linear functions through the convex combinationof weighting variables associated with function domain vertices. Theconvex combinations are limited to a single polytope by ensuring thatat most two and consecutive weighting variables are nonzero in eachaxis. The implementation of this model relies on specially ordered setsof variables of type II, the so-called SOS2 variables. Existing general-purpose solvers, such as CPLEX or Gurobi, provide native support forSOS2 variables. The SOS2 model is explicitly described for theillustrative function depicted in Fig. 2(a) and (b):

qinj X

pAKwhp;1500 1500p;2000 2000p;2500 2500p;3000 3000;

pwh X

qAKinj200;q 200250;q 250300;q 300350;q 350;

qliq X

pAKwhp;1500 ~f liqp;1500p;2000 ~f liqp;2000p;2500

~f liqp;2500p;3000 ~f liqp;3000;

1X

pAKwhp;1500p;2000p;2500p;3000;

p;qZ0; 8p; qAKwh Kinj;

pwh p;1500p;2000p;2500p;3000; 8pAKwh;

qinj 200;q250;q300;q350;q; 8qAKinj;

pwhpAKwh and qinjqAKinj are SOS2;

where ~f liqqinj;pwh is the well production function, Kwh f200;250;300;350g is the set of well-head pressure breakpoints,and Kinj f1500;2000;2500;3000g is the set of lift-gas breakpoints.Since pwhpAKwh is SOS2, only two consecutive variables can benonzero, let us say 250wh and

300wh . Likewise, suppose that

1500inj and

2000inj are nonzero. Consequently, only the weighting variables 250;1500,250;2000, 300;1500 and 300;2000 can be nonzero andmust add up to one.Therefore, pwh; qinj is constrained to be inside the polyhedron250;300 1500;2000.

Appendix B. Equations

The well-production functions bqnliq are approximated in theMILP formulation with the following equations:

For all nAN :~qnoil

Xqi ;pkAKn

nqi ;pk bqnliqqi; pk 1WCUTn; B:1a~qngas

Xqi ;pkAKn

nqi ;pk bqnliqqi; pk GLRnqninj; B:1b~qnwater

Xqi ;pkAKn

nqi ;pk bqnliqqi; pk WCUTn; B:1cqninj

Xqi ;pkAKn

nqi ;pk qi; B:1d

pnwhrX

qi ;pkAKnnqi ;pk pkp

n;maxwh 1yn; B:1e

pnwhZX

qi ;pkAKnnqi ;pk pkp

n;maxwh 1yn; B:1f

yn X

qi ;pkAKnnqi ;pk ; B:1g

nqi ;pkZ0; 8qi; pkAKn; B:1h

nqi X

pkAKnpnqi ;pk ; 8qiAK

ni ; B:1i

npk X

qiAKninqi ;pk ; 8pkAK

np; B:1j

nqi qiAKni ; npkpkAKnp are SOS2: B:1k

The pressure drops dpn;mjp qn;mliq ;GORn;m; pnds in the jumpers areapproximated in the MILP formulation by

For all nAN ;mAMn:~pn;mjp

Xql ;gor;pdsARn;m

n;mql ;gor;pds dpn;mjp ql; gor; pds; B:2aqn;moil

Xql ;gor;pdsARn;m

n;mql ;gor;pds ql 1WCUTn; B:2b

qn;mgas X

ql ;gor;pdsARn;mn;mql ;gor;pds ql 1WCUT

n gor; B:2c

qn;mwater X

ql ;gor;pdsARn;mn;mql ;gor;pds ql WCUT

n; B:2d

pndsrX

ql ;gor;pdsARn;mn;mql ;gor;pds pdsp

n;maxds 1zn;m; B:2e

Table 6Computational results.

Gas-lift Routing Oil production Solving statistics

Time Nodes Gap(s) (%)

High Automatic 2561.82 82 12 490 0.00Single 2523.53 140 20 836 0.00

Medium Automatic 2556.97 1800 470 867 0.20Single 2506.64 106 19 336 0.00

Low Automatic 2523.61 1800 196 403 1.52Single 2482.76 36 18 146 0.00


pndsZX

ql ;gor;pdsARn;mn;mql ;gor;pds pdsp

n;maxds 1zn;m; B:2f

GLRnX

qi ;pkAKnnqi ;pk

qibqnliqqi; pkrGLRn;max1zn;m

Xql ;gor;pdsARn;m

n;mql ;gor;pds gor 1WCUTn; B:2g

GLRnX

qi ;pkAKnnqi ;pk

qibqnliqqi; pkZGLRn;max1zn;m

Xql ;gor;pdsARn;m

n;mql ;gor;pds gor 1WCUTn; B:2h

zn;m X

ql ;gor;pdsARn;mn;mql ;gor;pds ; B:2i

n;mql ;gor;pdsZ0; 8ql; gor; pdsARn;m; B:2j

zn;mryn; 8mAMn; B:2k

n;mql X

gorARn;mg

XpdsARn;mp

n;mql ;gor;pds ; 8qlARn;ml ; B:2l

n;mgor X

qlARn;ml

XpdsARn;mp

n;mql ;gor;pds ; 8gorARn;mg ; B:2m

n;mpds X

qlARn;ml

XgorARn;mg

n;mql ;gor;pds ; 8pdsARn;mp ; B:2n

n;mql qlARn;ml ; n;mgor gorARn;mg ; and

n;mpds

pdsARn;mp are SOS2: B:2o

Appendix C. Nomenclature

The nomenclatures for sets, parameters, variables, and func-tions appear in Tables C1, C2, C3, and C4, respectively.

Table C1Sets.

Sets Description

N Set of wellsNm Subset of wells that can send production to manifold m: NmDNM Set of manifoldsMn Subset of manifolds receiving production from well n:MnDMH Set of phase ows: Hfoil, gas, and watergKni Gas-lift breakpointsKnp Well-head pressure breakpointsKn Breakpoints for approximating bqnliq: Kni KnpRn;ml Liquid ow rate breakpoints for jumpersRn;mg Gasoil ratio breakpoints for jumpersRn;mp Pressure downstream the choke breakpoints for jumpersRn;m Breakpoints for approximating cpn;mjp : Rn;ml Rn;mg Rn;mpQml Liquid rate breakpoints for the owlinesQmg Gasoil ratio breakpoints for the owlinesQmw Water-cut breakpoints for the owlinesQm Breakpoints for approximating cpm: Qml Qmg Qmw

Table C2Parameters.

Parameter Description

qn;L Vector with lower bounds on well n ow for all phases hAHqn;U Vector with upper bounds on well n ow for all phases hAHpn;maxwh Big-M value for well-head pressure

pn;maxds Big-M value pressure downstream the choke

pn;maxhipps Bound provided by the HIPPS

GLRn Gasliquid ratio for well nWCUTn Water cut for well nGLRn;m;max Maximum gasliquid ratio for jumper (n;m)qm;maxh Maximum value for the ow of phase h in manifold m

qm;max Vector with the maximum ows for all phases: qm;maxh : hAHqmaxg Gas compression capacity in the platformqmaxl Liquid handling capacity in the platformqmaxw Water treatment capacity in the platformqmaxinj Limit for gas-lift injection

pm;S Nominal pressure at the separator

Table C3Variables.

Variable Description

~qnh Flow of phase hAH produced by well n~qn Vector with all phase ows produced by well n: ~qnh : hAHqn;mh Flow of phase hAH sent by well n to manifold mqn;m Phase ow vector from well n to manifold m: qn;mh : hAHqmh Total ow of phase hAH received by manifold mqm Vector with all phase ows received by manifold m: qmh : hAHqninj Pressurized gas rate injected in well n

pnwh Well-head pressure of well npnds Pressure downstream the production choke of well nnqinj ;pwh Weighting variable for the PWL approximation of bqnoilyn Binary variable indicating whether well n is producingymman Binary variable indicating whether manifold m is producingnqi SOS2 variable on gas-lift to approximate bqnliqnpk SOS2 variable on well-head pressure to approximate bqnliqn;mql ;gor;pds Weighting variable for the PWL approximation of cpn;mjp~pn;mjp PWL approximation of the pressure drop function cpn;mjpGLRn;m Gasliquid ratio of ows in the jumperszn;m Binary variable indicating if well n is producing to manifold mn;mql SOS2 variable for jumper oil ow rate breakpoints

n;mgor SOS2 variable for jumper gasoil ratio breakpoints

n;mpds SOS2 variable for pressure downstream the choke breakpoints

~pm PWL approximation of the pressure drop function cpmpm Manifold pressureGORm Gasoil ratio of uids received by manifold mmqliq ;gor;wcut Weighting variable for the PWL approximation of cpmmqliq SOS2 variable for the owline liquid ow rate breakpoints

gorm SOS2 variable for owline gasoil ratio breakpointswcutm SOS2 variable for owline water-cut breakpoints

Table C4Functions.

Function Description

f Objective function expressed as the total oilproducedbqnliqqninj; pnwh Well production function of well ncpn;mjp qn;mliq ;GORn;m ;pnds Pressure-drop function of the well-manifold jumper(n,m)cpmqmliq ;GORm ;WCUTm Pressure drop in the owline of manifold m


References

Aguiar, M.A.S., Camponogara, E., Silva, T.L., 2014. A mixed-integer convex formula-tion for production optimization of gas-lifted oil wells with routing andpressure constraints. Braz. J. Chem. Eng. 31 (2), 439455.

Alarcn, G.A., Torres, C.F., Gmez, L.E., 2002. Global optimization of gas allocation toa group of wells in articial lift using nonlinear constrained programming.J. Energy Resour. Technol. 124 (4), 262268.

Beale, E.M.L., 1980. Branch and bound methods for numerical optimization of non-convex functions. Comput. Stat. 80, 1120.

Beale, E.M.L., Tomlin, J.A., 1970. Special facilities in a general mathematicalprogramming system for non-convex problems using ordered sets of variables.In: Proceedings of the Fifth International Conference on Operations Research,Tavistock, London, pp. 447454.

Buitrago, S., Rodrguez, E., Espin, D., 1996. Global optimization techniques in gasallocation for continuous ow gas lift systems. In: SPE Gas TechnologySymposium, Calgary, Canada.

Camponogara, E., de Conto, A., 2009. Lift-gas allocation under precedence con-straints: MILP formulation and computational analysis. IEEE Trans. Autom. Sci.Eng. 6 (July (3)), 544551.

Camponogara, E., Plucenio, A., Teixeira, A.F., Campos, S.R.V., 2010. An automationsystem for gas-lifted oil wells: model identication, control, and optimization.J. Pet. Sci. Eng. 70 (34), 157167.

Codas, A., Camponogara, E., 2012. Mixed-integer linear optimization for optimallift-gas allocation with well-separator routing. Eur. J. Oper. Res. 212, 222231.

Codas, A., Campos, S.R.V., Camponogara, E., Gunnerud, V., Sunjerga, S., 2012.Integrated production optimization of oil elds with pressure and routingconstraints: the Urucu eld. Comput. Chem. Eng. 46, 178189.

Fang, W.Y., Lo, K.K., 1996. A generalized well-management scheme for reservoirsimulation. SPE Reserv. Eng. 11 (2), 116120.

Fourer, R., Gay, D.M., Kernighan, B.W., 2002. AMPL: A Modeling Language forMathematical Programming. Duxbury Press, Pacic Grove, California, UnitedStates of America.

Gunnerud, V., Foss, B., 2010. Oil production optimizationa piecewise linear model,solved with two decomposition strategies. Comput. Chem. Eng. 34 (11),18031812.

Kosmidis, V., Perkins, J., Pistikopoulos, E., 2004. Optimization of well oil rateallocations in petroleum elds. Ind. Eng. Chem. Res. 43 (14), 35133527.

Kosmidis, V., Perkins, J., Pistikopoulos, E., 2005. A mixed integer optimizationformulation for the well scheduling problem on petroleum elds. Comput.Chem. Eng. 29 (7), 15231541.

Litvak, M., Darlow, B., 1995. Surface network and well tubinghead pressureconstraints in compositional simulation. In: Proceedings of the SPE ReservoirSimulation Symposium, San Antonio, TX.

Misener, R., Gounaris, C.E., Floudas, C.A., 2009. Global optimization of gas liftingoperations: a comparative study of piecewise linear formulations. Ind. Eng.Chem. Res. 48 (13), 60986104.

Silva, T.L., Camponogara, E., 2014. A computational analysis of multidimensionalpiecewise-linear models with applications to oil production optimization. Eur.J. Oper. Res. 232 (3), 630642.

Silva, T.L., Codas, A., Camponogara, E., 2012. A computational analysis of convexcombination models for multidimensional piecewise-linear approximation inoil production optimization. In: Proceedings of the IFAC Workshop on Auto-matic Control in Offshore Oil and Gas Production, Trondheim, Norway, pp. 292298.

Tomlin, J.A., 1988. Special ordered sets and an application to gas supply operationsplanning. Math. Program. 42 (13), 6984.

Vielma, J.P., Ahmed, S., Nemhauser, G., 2010. Mixed-integer models for nonsepar-able piecewise-linear optimization: unifying framework and extensions. Oper.Res. 58 (MarchApril (2)), 303315.

Yeten, B., Brouwer, D., Durlofsky, L., Aziz, K., 2004. Decision analysis underuncertainty for smart well deployment. J. Pet. Sci. Eng. 44 (12), 175191.


Modeling of flow splitting for production optimization in offshore gas-lifted oil fields: Simulation validation and...IntroductionFlow splitting modeling and validation for production optimizationNonlinear modelPiecewise-linear approximationMILP approximation for splitting

Simulation-based validationSynthetic field modelingSimulation analysis

Application to production optimizationMixed-integer linear programming model

Model synthesis and simulation analysisThe gas-lifted oil production systemModel synthesisGrid-fitting procedureModel analysis

Computational analysisSummaryAcknowledgmentsSOS2 modelEquationsNomenclatureReferences

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