Prediction of Condensate-to-Gas Ratio for Retrograde GasCondensate Reservoirs Using Artificial Neural Network with ParticleSwarm OptimizationSohrab Zendehboudi,*,† Mohammad Ali Ahmadi,‡ Lesley James,§ and Ioannis Chatzis†

†Chemical Engineering Department, University of Waterloo, Waterloo, ON, Canada, N2L 3G1‡Faculty of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Khuzestan, Iran§Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada, A1B 3X5

ABSTRACT: Condensate-to-gas ratio (CGR) plays an important role in sales potential assessment of both gas and liquid,design of required surface processing facilities, reservoir characterization, and modeling of gas condensate reservoirs. Field workand laboratory determination of CGR is both time consuming and resource intensive. Developing a rapid and inexpensivetechnique to accurately estimate CGR is of great interest. An intelligent model is proposed in this paper based on a feed-forward artificial neural network (ANN) optimized by particle swarm optimization (PSO) technique. The PSO-ANN model wasevaluated using experimental data and some PVT data available in the literature. The model predictions were compared with fielddata, experimental data, and the CGR obtained from an empirical correlation. A good agreement was observed between thepredicted CGR values and the experimental and field data. Results of this study indicate that mixture molecular weight amonginput parameters selected for PSO-ANN has the greatest impact on CGR value, and the PSO-ANN is superior over conventionalneural networks and empirical correlations. The developed model has the ability to predict the CGR with high precision in a widerange of thermodynamic conditions. The proposed model can serve as a reliable tool for quick and inexpensive but effectiveassessment of CGR in the absence of adequate experimental or field data.

1. INTRODUCTIONGas condensates are highly valued hydrocarbon resources. Gascondensate reservoirs are characterized by their complicatedflow regimes and intricate thermodynamic behaviors.1−3 Multi-phase flow of both gas and liquid condensates is expected during theproduction of a gas condensate reservoir with dynamic phasebehavior from the porous rock to the surface facilities. The Arunfield in Indonesia, South Pars field in Iran, and Greentown field inthe USA with condensate-to-gas ratio (CGR) ranging from 10 to300 STB/MMSCF are very large gas condensate reservoirs in theworld.1−4 Condensate production due to pressure drop in thereservoir improves project economics through condensate sales asproduced liquid condensate is the only potential source of revenuein the severe case of a poor gas market.3−5

Gas condensate reservoirs differ from conventional gas reservoirsin their thermodynamic and flow behaviors. Prediction and optimi-zation of hydrocarbon recovery in this type of reservoirs requirecareful reservoir analysis, development plan, and project man-agement.1−4 Isothermal condensation of liquids lowers the gasdeliverability in the reservoir due to condensate blockage.4,6,7

Low gas relative permeability in the gas−liquid region near theproduction well is the main reason behind reduction of gas flowrate.4,6−8 Proper production strategy for gas condensate re-servoirs can minimize or even avoid condensate blockage, how-ever the prediction of key parameters like CGR is necessary.High accuracy in prediction methods is needed in order toprovide an overall acceptable estimate of the performance ofgas condensate reserves before the implementation of a propergas recovery plan. Numerical and analytical modeling, PVTstudies, and characterizations of retrograde gas condensate reservoirs

remain challenging to the reservoir engineers as a huge amountof time and energy is required to carry out such studies mentionedabove, efficiently.4−6,9,10

The phase behavior of a gas condensate reservoir is stronglydependent on the P-T envelope and thermodynamic conditionsof the hydrocarbon mixture. Proper thermodynamic knowledgeof phase behavior in this type of reservoirs is essential in orderto predict the recovery factor and CGR. Several approaches areavailable in the literature for performance analysis of retrogradegas condensate reservoirs.7−15 Thomas et al. (2009) proposed asystematic strategy for performance prediction of gas conden-sate reservoirs based on relative permeability and two-phasedynamic measurements.11

Although measuring nonlinear parameters such as CGR anddewpoint pressure is viable through some tests (e.g., ConstantComposition Expansion (CCE) and Constant Volume Deple-tion (CVD)), however, the laboratory and field tests are quitecostly and people and time intensive.16,17 As quick, inexpensiveand efficient alternatives, other techniques that includetheoretical or/and laboratory tests combined with artificialintelligent systems were suggested by a number of research-ers.18−21 Several models have been reported in the literature topredict flow and thermodynamic behaviors of gas condensatereservoirs.12,13,15,22−26 Clearly, prediction of production rate iscentral in design of wellhead equipment and surface facilities. Alaviet al. (2010) used a modified Peng-Robison equation of state in

Received: January 4, 2012Revised: April 18, 2012Published: April 22, 2012

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their model combined with some experiments to determinephysical properties of gas condensate mixtures. Their main goalwas developing an appropriate design to optimize the amountof the produced condensate in a gas reservoir when pressuredecreases below the dewpoint.15 Jokhio et al. (2002) employedthe Whitson and Fevang’s model to estimate the extent of producedliquid condensate.24,25 Al-Farhan et al. (2006) examined the changesin CGR at various process conditions with considering a three-stageseparation process optimized using a four-layer back propagation(BP) artificial neural network (ANN).26 A number of evolutionaryalgorithms such as genetic algorithm (GA), pruning algorithm, andback propagation were also used to determine the network struc-ture and its associated parameters (e.g., connection weights).26−32

England (2002) developed a global polynomial correlation forCGR in terms of P and T based on a PVT study under specificthermodynamic and geological conditions.33

Accurate determination of PVT properties such as CGR isnecessary to predict and more accurately history match produc-tion rates and recovery factors. Many correlations exist in theliterature based on PVT data for various types of hydrocarbonmixtures. Most of these correlations were obtained with linearor nonlinear multivariable regression or graphical techniqueswhere some are not very accurate.33−38 Artificial neural network(ANN) systems are potentially reliable tools for the predictionof PVT parameters such as CGR in gas condensate reservoirs ifthe networks are effectively trained. The enormous intercon-nections in the ANN offer a wider degree of freedom or a largenumber of fitting variables. Therefore, ANN is better able todescribe the nonlinearity of the process compared to classicalregression methods. Another advantage of artificial neural networksis that they are dynamically adaptive where they are able to learnand adjust to new conditions in which the efficiency of ANN is notadequate with old situations. In addition, the ANNmodel is capableof handling systems with multiple inputs and outputs.39,40

This paper highlights the economic significance, productioncharacteristics, production methods, and P-T behavior of gascondensate reservoirs with a brief description of commontechniques for determining the CGR. An intelligent model isdeveloped, based on a feed-forward artificial neural network(ANN) optimized by using particle swarm optimization (PSO),and examined to predict CGR in terms of dewpoint pressure,temperature and mixture molecular weight for retrograde gascondensate reservoirs through a rapid and accurate manner.PSO is employed to determine the initial weights of the param-eters used in the ANN model. The PSO-ANN model is testedusing extensive field and laboratory data. Results from thisstudy reveal that the developed model can predict the CGRwith high accuracy. The developed model can serve as a reliabletool for quick and cheap but efficient estimation of CGR in theabsence of sufficient laboratory or/and field data especiallyduring the initial stages (e.g., determination of number of wells,well placement, and estimation of optimum production rate) ofdevelopment of the gas condensate reservoirs.

2. RETROGRADE GAS CONDENSATE RESERVOIRSA typical P-T phase diagram for gas condensate reservoirs(Figure 1) has a critical temperature lower than the reservoirtemperature. However, its cricondentherm is larger than the re-servoir temperature. Gas condensate systems commonly appearas a single gas phase in the reservoir at the time of discovery. Asthe reservoir is produced, the reservoir pressure declines fromthe reservoir to the production well and to the surface facilities,resulting in gas condensation to produce a liquid phase. This

isothermal condensation due to the reduction of pressure belowthe dewpoint pressure of the original hydrocarbon mixture iscalled retrograde condensation.1−4,11

Gas condensate reservoirs generally produce in the range of30−300 STB/MMSCF (standard barrels of liquid per millionstandard cubic feet of gas).5,12,15,23,33−38,41−47 The depth ofmajority of the reported retrograde gas condensate fields rangesfrom 4000−10,000 ft (∼1200−3000 m).5,12,15,23,33−38,41−47

Also, the ranges of pressure (P) and temperature (T) for thistype of gas reservoirs are usually between 3000−8500 psi (20−55 MPa) and 150−400 °F (65−200 °C), respec-tively.5,12,15,23,33−38,41−47 Clearly, few gas condensate reservoirsmay be found out of the ranges reported above for T, P, anddepth as gas condensate reservoirs may have pressures lowerthan 2000 psi and temperatures under l00 °F.47 The pressureand temperature values along with the wide range ofcompositions result in gas condensate mixtures that exhibitdifferent and complex thermodynamic behaviors.1−4,10,11

Since the formation has a lower permeability to liquid andthe ratio of liquid viscosity to gas viscosity is fairly high in the gas-condensate reservoirs, the main portion of the condensed liquid inthe reservoir is unrecoverable and considered as a condensate loss.Condensate loss is a main concern in gas reservoir engineeringbecause the liquid condensate holds the more economicallyvaluable intermediate and heavier components of the originalhydrocarbons that are trapped in the reservoir.1−12

Characterizing gas-condensate systems is a complex task forresearchers and subject matter experts, since multiphase flow inthe formation and also variations of the fluid compositionsignificantly obscure the analysis of well tests. In the researcharea of gas-condensate reservoirs, the topics such as welldeliverability, PVT analysis, well test interpretation, multiphaseflow, rock characterization, relative permeability model, viscousstripping of condensate and interblock non-Darcy flow havebeen the common problems for long time.1−15

2.1. Three Flow Regimes. According to Fevang (1995),three main different flow regimes can be identified in thehydrocarbon mixture flowing toward a production well in a gas-condensate reservoir during depletion, as depicted in Figure 2:3,25

• Single phase gas (Region 3): A region that is far from theproduction well and has a pressure higher than the dewpointvalue and thus only single gas phase exists in the region.

Figure 1. A typical gas condensate phase envelope.

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• Condensate build-up (Region 2): A region where thereservoir pressure is below the dewpoint pressure. In thiscase, production of liquid condensate is observed in thereservoir. However, the condensate phase will not flowdue to its low saturation value. Consequently, the flowingphase in this region is still composed of just the single gasphase. The flowing gas loses the heavier components inthis region as the reservoir pressure decreases during theproduction process.

• Near well (Region 1): Region 1 is an inner near-well partin which the reservoir pressure is more below thedewpoint. In this region, saturation of the liquidcondensate exceeds the critical value and the condensatebuild-up turns out to be mobile. The gas phase mobilityremains considerably low due to the presence of liquidcondensate phase. In the near wellbore area (Region 1),two counter-acting forces dictate the degree to which thecondensate phase can move.3,25,48−50 Viscous forces,which depend on viscosity and velocity of the gas canstrip the condensate from the formation. The capillarypressure, which is a function of interfacial tension, isresponsible for keeping the condensate in the pores ofthe rock. Experimental studies suggest that relative per-meability and critical saturation of condensate change interms of capillary number.3,25,48−50 The ratio of viscousforces to capillary forces has a significant effect on pro-duction rate when the values of condensate velocity andcondensate saturation are fairly high. As the relative per-meability of condensate phase is enhanced at greater gasvelocities in this region, the critical saturation of condensatedecreases significantly.3,25,48−50 Another important viewpointhere is that stripping by viscous forces affects condensateaccumulation in Region 1, and this causes a lower liquidsaturation around the wellbore.3,25,48−50

2.1.1. Condensate Blockage. Gas mobility has themaximum value further away from the wellbore where reservoirconditions dominate and gas saturation is 100 % as shown inFigure 2, Region 3. Gas mobility declines significantly in Region 2where condensate accumulates and lower gas saturation is

observed. Region 1 shows an even lower gas saturation andhigher liquid condensate saturation such that the liquid con-densate is mobilized. Even if the condensate production is negligible,the gas mobility and consequently the well deliverability remain con-siderably low. This phenomenon is known as “condensateblockage”.3,25,41 The major resistance for gas to flow is inRegion 1, and the blockage influence depends mostly on therelative permeability of the gas phase and the volume of Region 1.The gas flow resistance in Region 2 is much lower than thatin Region 1.

2.1.2. Hydrocarbon Recovery and Composition Change.Composition of the gas varies as it moves toward Region 2 andthe flowing gas becomes leaner of intermediate and heavy com-ponents (e.g., C4

+) in the reservoir.3,25,41 Hence, the heaviercomponents build up in Regions 1 and 2 as the reservoir pres-sure declines. Overall, the composition of the mixture every-where in Region 1 or 2 contains heavier components when thereservoir pressure drops below the dewpoint. Region 1, nearwellbore with significant liquid condensate saturation, canrapidly develop during production. This will result in produc-ing leaner gas compared to the initial gas composition andleaving the intermediate and heavier components in Regions 1and 2.3,25,41

2.2. Production Methods and Pressure Maintenance.Production and/or recovery methods for gas condensatereservoirs usually include fast depletion, slow depletion, fulldry gas cycling, partial dry gas cycling, seasonal dry gas cycling,noncondensable gas (NCG) injection, injection of tuned gascomposition, and water injection.3,25,41−43 Most of thesemethods are employed for the purpose of pressure main-tenance. In general, pressure maintenance processes areclassified in four main groups: 1) an active water drive aftermodest decline of pressure from early production, 2) waterinjection techniques, 3) gas injection, or (4) combinations ofthese.3,25,41−43 As times goes on, some gas condensatereservoirs may experience thermodynamic conditions close totheir critical points. In this case, reservoirs appear to be goodcandidates for some recovery methods such as injection oftuned gas compositions to offer miscibility and phase-transformprocesses that can enhance the recovery factor.3,25,41−43

Figure 2. Three regions of condensate and gas flow behavior in a typical gas condensate reservoir (modified from Roussennac (2001)41).

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2.2.1. Depletion. Natural depletion in the form of fast andslow depletions involves just production. No dry or residue gasis recycled to the reservoir. Condensate recovery is effective justat the beginning of production when the produced mixtureholds all original components of liquefiable hydrocar-bons.3,25,41−43 The reservoir pressure is high enough (greaterthan the dewpoint pressure) as there is only gas phase in theformation at this stage, referred to as “fast depletion”. As thereservoir produces hydrocarbons and the pressure drops belowthe dewpoint, the retrograde condensation process starts. Theheavier components of hydrocarbons mixture condense in thereservoir. The percentage of condensate phase is variable anddepends on the composition of the mixture.3,25,41−43 Recoveryof the liquid condensates at lower pressures is not guaranteed.Below the dewpoint, the relative permeability of gas phasedeclines causing slow depletion of the fluids. Methods havebeen developed in order to maintain pressure, enhance therecovery, and finally produce condensate at the surface.Advantages and disadvantages of depletion processes includethe following:3,25,41−43

Advantages:

• High recovery factors for gas under solution gas regimes• Low cost for reservoir development• Reservoir studies with fairly low cost needed

Disadvantages:

• Low liquid recovery efficiencies and almost irreversible lossof retrograde condensate which is left in the formation

• Reduction in well production rates when reservoirpressure declines and retrograde condensation happensin the bottom hole

• Extra compression needed as reservoir pressure is reduced• Difficulty in managing the production process when

there is an active water drive in the reservoir• Difficulty in managing production wells operating under

notably declined reservoir pressure

2.2.2. Water Injection. Only a few gas condensate reservoirsare reported to operate under natural water drive (aquiferexpansion).3,25,41−43 A strong water drive is needed in order tomaintain high enough pressure to reduce condensates fromforming and be economical. Therefore, a careful process designand an appropriate economic analysis should be implemented ifa water injection process is selected as a production method oras a technique for pressure maintenance.3,25,41−43 Otherimportant aspects such as displacement efficiency of waterpushing gas, bypassing potential, and loss of condensate phaseshould be considered when water breakthrough occurs viapermeable zones. The mobility ratio for water flooding issatisfactorily low, leading to attaining high areal sweepefficiencies.3,25,41−43 The presence of layers with very differentpermeabilities in a formation may cause early water break-through in the production well, resulting in prematureabandonment of the production plan.3,25,41−43 In general, thefollowing concerns are present while employing water injectionmethod for gas condensate reservoirs:3,25,41−43

• The utilization of water injection in a massivelyheterogeneous gas-condensate reservoir is doubtful tobe possible technically and economically due to earlybreakthrough and consequently loss of productivity.

• Up to 50% of the gas might be bypassed and trapped inthe formation.

• It may not be possible to mobilize the gas previouslytrapped in the formation through a subsequent depressu-rization stage.

• Well lift could be a serious problem if high water cutsbefore and during the blowdown are experienced.

2.2.3. Gas Cycling. In dry gas injection processes, the goal isto keep the reservoir pressure over or near the dewpoint inorder to minimize the amount of condensate.3,25,41−43 Dry gascycling is considered as a particular type of miscibledisplacement to improve recovery.3,25,41−43 This method isvery efficient in terms of displacement efficiency and havingeconomical condensate production rates, especially when thereare no markets available for the produced gas. There are twomain groups for dry gas cycling, namely full cycling and partialcycling.3,25,41−43 Full gas cycling is when all the dry gasseparated at the surface is recycled into the formation. Such acycling generally leads to a material balance deficiency aspressure may slowly reduce. Some dry gas can be added asmake-up gas to the injection preventing this occurrence. Partialcycling provides lower pressure maintenance over the dewpointthan the full gas cycling method.3,25,41−43 A gas sales contractusually dictates the amount of dry gas for injection into theformation where any extra gas produced is re-injected. Forexample, the partial cycling in the form of a seasonal gasinjection is able to provide more dry gas over the off-peaksummer months if the reservoir can still be operated at itsmaximum production rate.3,25,41−43 Besides lowering the loss ofcondensate in the reservoir, the pressure maintenance resultingfrom gas cycling hinders water influx and the entrapment of wetgas with its restrained condensate in the rear of the advancingwater front.3,25,41−43 The gas cycling process ends when gasbreakthrough into the production wells is observed and the rateof condensate production is not economical. Dry gas cyclingmethods should be considered in terms of important featuressuch as mobility ratio, heterogeneity, and gravity which mostlyaffect the recovery factor.3,25,41−43 The demand for dry gasgenerally controls the amount of dry gas available for gascycling. Therefore, the main disadvantage of dry gas cycling isthat the sale of produced gas should be postponed untilcondensate production declines to its economical limit. The useof inert gas to compensate the amount of gas required for gascycling process is an economical option for pressure main-tenance and recovery enhancement.3,25,41−43 The inert gas ornoncondensable gas injection generally uses nitrogen, carbondioxide, and combustion or flue gases. Nitrogen is the mostuseful gas for this type of cycling since it is noncorrosive and hasa high compressibility factor, leading to requiring less volume.The most important factors affecting economic viability of a

gas-cycling project are liquid content of reservoir, productprices, and costs of makeup gas. The utilization of inert gascycling has some advantages and disadvantages. The key benefitis that inert gas cycling permits early sale of gas and condensatephases, leading to higher net income.3,25,41−43 In addition, agreater recovery of total hydrocarbons is obtained since theformation holds large amounts of nitrogen rather thanhydrocarbon gas when the reservoir is abandoned. Operationalproblems and higher operating costs are the result of corrosionif combustion or flue gas is employed for the cycling process.Extra capital costs are needed to separate the inert gas from thesales gas, to pretreat prior to compression, and to providesurface reinjection facilities. Also, premature breakthrough ofinert gas due to high degrees of reservoir heterogeneity causes

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additional operating expenses to find marketable gas sales.Some other shortcomings for all types of gas cycling processesinclude the following: 1) Gas injection is an expensive process; 2)It takes too long to reach the ultimate recovery of the field; 3)Production rate and recovery factor of gas are fairly low; therefore,gas production is less beneficial; 4) Heavy condensate fractions ofhydrocarbons are precipitated in bottom hole regions; 5)Economic efficiency is poor when active water drive happens;and 6) Reservoir studies are difficult and costly.3,25,41−43

3. CGR DETERMINATION TECHNIQUESThe value of CGR in a gas-condensate reservoir depends onvarious factors such as API gravity, gas composition, pressure,and temperature. The preference is to determine CGR usingdetailed compositional tests if adequate experimental facilitiesand samples are available and also CGR values with highaccuracy are required. However, it is usually common in PVTstudies to develop CGR empirical correlations, equation ofstate (EOS), and numerical models due to the large number ofparameters and the complexity of the experiments to obtainCGR. It should be noted here that empirical and theoreticaltechniques such EOS and intelligent methods usually havedifficulties to predict thermodynamic properties around criticalpoint with high precision. Therefore, the accuracy of availableempirical relationships strongly depends on the quality of thesamples and the proximity of reservoir conditions (e.g., P and T)to the fluid critical point.41−43 In this section, some of thetechniques used for CGR determination are described, briefly.3.1. Laboratory Methods. There are two main exper-

imental methodologies for measuring CGR, namely constantcomposition expansion (CCE) and constant volume depletion(CVD). CCE is primarily used to determine the dewpoint ofhydrocarbon mixtures and also CGR. In addition, the CVDmethod is conducted in order to simulate the behavior ofreservoir production. Flash vaporization is another name forCCE and is simulated by expanding a mixture with a certaincomposition (zi) in sequence steps of pressure reduction. Thetechnical reader is encouraged to read refs 16 and 17 for a moredetailed description of the experimental procedures.3.2. Empirical Correlations. Several empirical correlations

were proposed for CGR prediction in the literature.33−38,44

Some field data such as reservoir pressure (P), reservoir tem-perature (T), and fluids specific gravity are needed to developPVT empirical correlations. For instance, Cho et al. (1985)developed an empirical correlation to estimate maximum CGRvalue for retrograde gas-condensate reservoirs.44 Their correlationis expressed in terms of reservoir temperature and the molefraction of heptanes-plus. Reliable experimental data are favoredversus the results obtained from the correlations when assessinggas-condensate reservoirs.33−38,44 Nevertheless, accurate exper-imental data often are not available. One advantage of correlationsis being helpful when only modest experimental information isavailable.3.3. Analytical Methods. Analytical methods are usually

based on EOS to evaluate CGR for gas-condensate reservoirs.This technique is very efficient to describe the phase behavior ofwell characterized mixtures at particular temperatures. However,the EOS should be tuned for each mixture composition and haspredictive potential limitations for new compositions. Therefore,an EOS method requires a detailed description of the mixture’scomposition. Obtaining such quantities is expensive and time-consuming. Furthermore, there is no universal EOS available forhydrocarbon mixtures, which is able to predict CGR and other

thermodynamic parameters over wide ranges of compositions,temperatures, and pressures with an acceptable accuracy. Some ofthe available analytical methods are including EOS models,material balance methods, and PVT delumping techni-ques.9,42,43,51−56

3.4. Numerical Modeling. Numerical simulations areanother tool to predict CGR. An accurate numerical methodto compute the CGR and well productivity is a fine-grid system.A fine-grid model has the ability to consider high-velocityeffects, and most commercial simulators now incorporateoptions to consider non-Darcy flow and complicated phasebehavior in different regions of gas-condensate reservoirs.Although, numerical simulations are proper for forecasting ofreservoir behavior in detail, simpler and faster engineeringcalculations look more appropriate for some applications such asCGR prediction.42,43 For instance, the book titled ‘The Practiceof Reservoir Engineering’ presents some simple theoreticalequations and forecasting methods to compute CGR value andother important parameters (i.e., sweep efficiency, relativepermeability and two-phase Z-factor) in gas condensatereservoirs when the gas cycling technique is employed.43 Itshould be also noted here that a strong understanding of thecondensate banking behavior in gas condensate reservoirs isrequired for analysis of well test data and prediction of pro-duction performance using numerical methods.7,8,10,22,41−43

4. EXPERIMENTAL PROCEDURE AND DATACOLLECTION

The CGR of the hydrocarbon mixtures from the South Pars/NorthDame gas field in Iran was measured by CCE and CVD methods andan average value of CGR was used for the intelligent model developedin this paper. The South Pars natural gas-condensate field is located inthe Persian Gulf. It is the world’s largest single gas field, sharedbetween Iran and Qatar. The field is composed of two independentgas-bearing formations, namely Kangan and Upper Dalan. Major gasreservoirs in this field mainly consist of limestone and dolomite. Bothporosity (ϕ) and permeability (k) vary widely in the field. For instance,the porosity varies from 0−35% and the permeability from zero to 1000mD in some regions of this gigantic gas field. In addition to the datagenerated from the experimental part, some data from the literature werealso used in developing the PSO-ANN model for CGR prediction. A partof the data used in this study is presented in Appendix A, and the rest canbe tracked in the open literature.5,12,15,23,33−38,41−85

5. ARTIFICIAL NEURAL NETWORKS THEORYInitially, McCulloch and Pitts (1943) developed the artificialneural network (ANN) theory as a mathematical technique.86

ANNs are biological inspirations on the basis of different brainfunctionality features.39,40,86 In recent years, ANN has beenbroadly employed for numerous applications such as processcontrol, behavior prediction, model recognition, systemclassification, communication, and vision aspects. In this theory,ANN is generally trained to solve nonlinear and complexproblems whose mathematical modeling appears to be adifficult job.39,40,86 The ANNs can take some information of acomplicated system and provide an adequate model even whenthe data are noise infected and incomplete. The objective of anANN model is to correlate a set of input data to a set of outputpatterns in a process. The network carries out this mappingscheme through training from some previous examples and alsodefining the input and output parameters for a particularsystem. ANNs contain many computational sections joinedtogether in a fairly complicated parallel structure. In the pro-cessing units, there are a great number of simple components

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called neurons that behave like axons to obtain an empiricalcorrelation between the inputs and outputs of a given process.39,40,86

Figure 3 presents a classical interconnected neural network inthe form of a multiple-layers structure. The network is com-

posed of one input layer, one hidden layer, and one outputlayer with various duties. Each linking line holds a certainweight. ANNs are trained via adjustment of the connectionweights as the desired values for output parameters areestimated with an acceptable accuracy.30,31,86−89 The mainfeature of a neural network is the training stage according to aset of numerical magnitudes measured in the modeling process.The precision of the model representative is stronglydependent on the topology (structure) type of the neuralnetwork.30,31,86−89 There are a number of different neuralnetwork structures and learning algorithms for the ANNtechnique. The back-propagation (BP) learning algorithm isrecognized as one of the most general algorithms. This learningalgorithm was selected in this study to predict the magnitude ofcondensate-to-gas ratio (CGR).30,31 BP is a multiple-layer feed-forward network that includes hidden layers linking the inputparameters to the output.30,31,39,40 The simplest accomplish-ment of back-propagation (BP) training is the biases and neuralnetwork weights updates toward negative gradients.30,31,39,40

The ANNs generally try to identify the data patternsthroughout the training process. ANNs obtain the flexibilityduring the process letting the analyst operator accomplishtesting the data patterns even if the work environment in themodel is changed. Although ANN might take some time tolearn an impulsive alteration, it is exceptional to adjustcontinuously varying information.30,31,86−89

6. PARTICLE SWARM OPTIMIZATIONParticle swarm optimization (PSO) is a global optimizationalgorithm for the problems in which a point or surface in amultidimensional space represents the best solution.90,91 Asshown in Figure 4, PSO is initialized with a set of randomparticles. Then, random positions and velocities are assigned tothe particles. After that, the algorithm starts searching foroptima through a number of iterations where the particles areflying in a hyperspace to find the potential solutions. As timegoes on, the particles learn to respond appropriately, based ontheir experience and also the experience of other particles intheir set.90−93 Every particle maintains the path of its bestposition that has obtained in the hyperspace to date.90,91 Thevalue of best position is named personal best. The overall best

magnitude of position achieved by any particle in thepopulation is called global best. In the period of iterationprocess, every particle experiences a tendency toward its ownpersonal best and also in the direction of the global best. This isdetermined through calculation of a new velocity term for eachparticle, based on the distance from its personal best positionand also its global best position. Random weights to personalbest and global position velocities are then assigned to gene-rate the new value for the particle velocity that shows effects onthe next position of the particle in the next iteration.90−93

Particle fitness mentioned in Figure 4 is a measure of how goodthe solution characterized by position is. For minimization pro-blems that are the most typical categories of cases solved byPSO, lower magnitudes of the fitness field are better thanhigher values. However, greater values of the fitness aresuperior for maximization problems.The position of a particular particle in a PSO-ANN model

indicates a series of weights for the current iteration.90,91 Thedimension of each particle is equal to the number of weightscorresponding to the network. Using mean squared error(MSE), the learning error of the network is calculated. Themean squared error of the network (MSEnet) is defined as

∑ ∑= −= =

MSE Y k T k12

[ ( ) ( )]net

k

G

j

m

j j1 1

2

(1)

where m is the number of output nodes, G is the number oftraining samples, Yj(k) is the expected output, and Tj(k) is theactual output. WhenMSEnet approaches to zero, the error of ournetwork would decrease.The most important advantage of the PSO approach in

comparison to other global minimization techniques likesimulated annealing is that many members contributing inthe PSO model cause the technique remarkably flexible as thePSO method prevents the neural network model from beingstuck on local minima.90−93 It should be also noted here thatthe particle swarm optimization (PSO) model is similar to thegenetic algorithm (GA) model; however, the PSO employs acollaborative approach rather than a competitive one.

6.1. Training Using Particle Swarm Optimization.Given a fixed architecture of ANN, all weights in connections

Figure 3. Architecture of three layers in an ANN model.30,31

Figure 4. A basic procedure for the PSO approach.

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can be coded as a genotype and applied in the PSO algorithmin order to train the network where the fitness function shouldbe the mean squared error of the network.90,91 Using validationand testing sets, some changes can be made to obtain betterfitness magnitudes with more generalization characteristics. Totrain the network by the particle swarm optimization, thefollowing equations are used90−93

ω= + − + −+V V c r P X c r P X( ) ( )it

it

1 it

it

it

2 2t

gt

it1

(2)

= ++ +X X Vit

it

it1 1

(3)

where Vit is defined as the velocity vector at iteration t, r1 and r2

indicate random numbers in the range [0,1], Pgt represents the

best ever particle position of particle i, and Pit refers to the

global best position in the swarm until iteration t.90,91 Otherparameters in the above equations are problem dependentvariables. For instance, c1 and c2 denote ‘‘trust’’ parametersindicating how much confidence the current particle c1 as acognitive parameter has in itself and how much confidence c2(social parameter) has in the swarm. ω represents the inertiaweight. The equations describe how the particle velocity valueand location are varying in an iteration procedure.90,91 Duringthe training process, the above equations will customize thenetwork weights until a stopping condition is satisfied. In thiscase, a lower MSE is achieved; however, a number of maximumiterations is used to end the algorithm if the examined stoppingcondition does not lead to termination in an appropriate time.Generally, the inertia weight (ω) decreases linearly with the

iteration number as follows90−93

ω ωω ω

= −−⎛

⎝⎜⎞⎠⎟t

ttmax

max min

max (4)

where ωmin and ωmax are the initial and final values of the inertiaweight, respectively, t is the current iteration number, and tmax isthe maximum number of iterations used in PSO. The values ofωmax = 0.9 and ωmin = 0.4 are the proper value throughempirical studies.90,91

6.2. PSO-ANN Model Development. A variety of modelswere calibrated to find the optimum configuration of the PSO-ANN algorithm. A PSO-ANN structure with appropriateparameters can exhibit a good performance.90−93 To design aproper network, six (6) parameters were studied as given below:a) Acceleration constant for global best (c1), b) Acceleration

constant for personal best (c2), c) Time interval, d) Number ofparticles, e) Number of maximum iterations corresponding to alow MSE that refers to the stopping condition, and f) Numberof hidden neuronsThe constant for global best (c1), constant for personal best

(c2), time interval, and number of particles have significanteffects on the optimal configuration of the PSO-ANNmodel.90−93 In addition, other parameters namely, maximumiteration, number of neurons in the hidden layer, and length ofinput and output data, are examined for calibration andoptimization of the model.The objective function employed here is mean squared error

(MSE). The optimization step is vital to make sure that MSE ortraining error is decreasing with an increase in the number ofiterations. Performance of the PSO-ANN is assessed usingcoefficient of determination (R2) and Nash-Sutcliffe coefficient(E2). If the values of R2 and E2 approach 1, the selectednetwork would be a perfect model to predict the systembehavior. The corresponding formulas for the above statistical

parameters and other two performance measures (e.g., meanabsolute error (MEAE) and maximum absolute error (MAAE))are presented in Appendix B. It should be noted here that asMEAE and MAAE become lower, better performance for thedeveloped neural networks is expected.To calibrate the PSO-ANN model, three layers were used in

this study. The number of input neurons is a function of thenumber of precursor data, and there is just one parameter (neuron)in the output layer. PSO-ANN was trained and then tested todetermine the best structure using different data sets as follows:

a) c1 and c2 values were changed from 1.4 to 2.4.b) The values of 0.005, 0.010, 0.015, 0.020, 0.025, and 0.030

were tested for time interval.c) Number of particles varied between 15 and 25 in the

training mechanism.d) Number of hidden neurons examined in this study was 3,

5, 7, and 10.e) Maximum iterations were examined between 300 and

600.

6.3. Selection of Input Variables. In general, ANN worksas a black box model that relates inputs variables and outputsparameters, which represents a mathematical relationship. It iscrucial that key physical parameters contributing in the outputof ANN models are used. Also, it is desirable to minimize thenumber of inputs for an ANN system. Selection of the bestinputs is based on the understanding of physics of the problem.As previously mentioned, experimental works, numericalmodeling, empirical equations, and PVT studies of gascondensate reservoirs have proved that CGR is a function oftemperature, pressure, and composition of fluid mixture in thereservoir. To take into account the effect of mixturecomposition, some of the published research studies usedmolecular weight of C7+.

41−43 In addition, there are somestudies in the open literature that considered molecular weightof C30+ in their computations.41−43,60−85 In order to select thebest inputs for the ANN method, the following sets of inputvariables were examined as shown in Table 1: 1) T, Pd,MwC30+; 2)T, Pd, MwC7+; 3) T, Pd, mixture composition, and 4) T, Pd, Mw ofthe mixture.

Input data set no. 3 that includes mole (or mass) fraction ofall components in the mixture seems appropriate for the PSO-ANN model; however, it makes the computation very time-consuming. On the other hand, it is difficult to construct such alarge data center in the neural network system, and many inputdata sometime cause errors in calculation of the desiredparameter. However, with input data set no. 4, we consider theeffect of mass or mole percentages of all components byselecting molecular weight and dewpoint pressure of themixture. As shown in Table 1, input data set no. 4 among theselected input data sets exhibits the best performance for thePSO-ANN.

Table 1. Performance of PSO-ANN for Various Input Dataof Variables

statisticalparameter

input data setno. 1

input data setno. 2

input data setno. 3

input data setno. 4

R2 0.8995 0.9108 0.9447 0.9705E2 0.8859 0.9036 0.9355 0.9548MSE 0.0976 0.0777 0.0541 0.0438MEAE (%) 12.1133 11.2461 7.7733 6.0004MAAE (%) 15.6432 13.5488 9.0064 8.1192

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6.4. Data and Model Evaluation. Availability of reliabledata is very important in the design of any ANN system. Thefirst step which should be considered in an ANN system isfinding parameters that affect the target parameter, consid-erably. CGR is a strong function of reservoir fluids properties. It isclear that reservoir characterizations (e.g., depth and permeability)dictate pressure distribution, local compositions, and temperaturein gas condensate reservoirs that affect instantaneous values ofCGR. Considering effects of reservoir characteristics, three mainparameters that have direct influences on the amount of CGRinclude the dewpoint pressure (Pd), the reservoir temperature(T), and molecular weight of mixture (Mw).

10−13,18−25

A set of 295 experimental data was collected to develop andexamine the proposed PSO-ANN model. The data seriesincludes 45 data from South Pars gas-condensate reservoir and250 data from the literature.5,12,15,23,33−38,41−85 The ranges ofthe parameters used to develop the PSO-ANN are given inTable 2.

It should be noted here that Table 2 covers wide ranges ofimportant parameters contributing in CGR. The above rangesare applicable for the most of real gas condensate casesencountered. If a data point is selected very far from the aboveintervals, the methodology to implement the PSO-ANN doesnot change. However, some parameters of the PSO-ANN suchas number of hidden neurons and number of particles shouldbe adjusted in a way that an acceptable match is obtained.Another alternative for such a case is adding a wavelet functionto the neural network structure in order to increase accuracy ofthe predictions.88−93

7. RESULTS AND DISCUSSIONPrecise determination of the CGR and prediction of itsvariations are central in developing a proper PVT model forutilization in reservoir engineering simulators. Development ofan artificial neural network (ANN) is discussed here which wasused to build a model to predict the amount of condensate-to-gas ratio. The ANN model trained with back-propagationnetwork employed the Levenberg−Marquardt algorithm forprediction of the CGR. The transfer functions in hidden andoutput layers were sigmoid and linear, respectively. PSO wasused as a neural network optimization algorithm, and the meansquare error (MSE) was used as a cost function in this

algorithm. The goal from implementation of the PSO was tominimize the cost function. Every weight in the network wasinitially set in the range of [−1, 1] and every initial particle wasa set of weights generated randomly in the range of [−1, 1]. Forexample, the initial and final values of the inertia (ωmin andωmax) were 0.4 and 0.5, respectively; r1 and r2 were also tworandom numbers in the range of [0, 1].The PSO-ANN model was evaluated with 3, 5, 7, 8, and 10

hidden neurons in the hidden layer. The results obtained usingdifferent numbers of hidden neurons are shown in Table 3. Theaccuracy of the simulations was increased with increase in thenumber of hidden neurons from 3 to 8. At 3, 5, and 7 hiddenneurons, the network did not appear to have enough degrees offreedom to learn the process accurately. However, theperformance of PSO-ANN began to decline at 10 hiddenneurons. Therefore, the optimum value for hidden neurons is 8in the PSO-ANN model, because it takes a long time for thePSO-ANN system to be trained at 10 hidden neurons and themodel may over fit the data.The acceleration constants (c1 and c2) correspond to the

stochastic acceleration that pushes each particle towardpersonal best and global best positions.91−93 The c1 constantdetermines the effect of the global best solution over theparticle, while the c2 constant describes how much the effect ofpersonal best solution has over the particle. Reasonable resultswere obtained when the PSO-ANN was trained withacceleration constants (c1 and c2) in the range of 1.4−2.4.Performance of the PSO-ANN was improved as the values of c1and c2 increased from 1.4 to 2.0. The main reason is that theparticles are attracted toward personal best and global best withhigher acceleration and abrupt movements when c1 and c2values experience an increase from 1.4 to 2.0. Afterward, thePSO-ANN performance reduced as c1 and c2 values increasedfrom 2.2 to 2.4. This trend indicates that the PSO-ANN is notable to converge with higher magnitudes of accelerationconstants. Thus, the best c1 and c2 values required for anoptimal structure of the PSO-ANN model were 2 where R2 andE2 were determined to be 0.9789 and 0.9581 for the trainingand 0.9087 and 0. 8563 for the testing process, respectively. Inaddition, the minimum values of MSE, MEAE, and MAAE wereobtained for both training and testing processes at c1 and c2values equal to 2. The results for the PSO-ANN trained withdifferent values of c1 and c2 are listed in Table 4.Number of particles in the PSO-ANN dictates the quantity of

the space covered in the problem. Table 5 shows the effect ofnumber of particles on the performance of the developedintelligent model. As seen in Table 5, the PSO-ANNperformance increases with an increase in the number ofparticles from 15 to 21. The developed neural network modelwas not able to describe well the data behavior at numbers ofparticles equal to 15 and 18 because the space covered in theproblem was not adequate for these numbers. Then, a decreasein the PSO-ANN was observed when the number of particles

Table 2. Ranges of the Parameters Used to Estimate theCGR

property min max

input temperature (T), °F 152 355dewpoint pressure (Pd), psi 2950 8642molecular weight (Mw), g/mol 16.5 44

output condensate-to-gas ratio (CGR), STB/MMSCF 4.5 293

Table 3. Performance of PSO-ANN Based on the Number of Hidden Neurons

training testing

number of hidden neurons R2 E2 MSE MEAE (%) MAAE (%) R2 E2 MSE MEAE (%) MAAE (%)

3 0.8253 0.8244 0.0823 8.6542 13.7561 0.7520 0.7659 0.0998 10.4588 15.78655 0.8724 0.8609 0.0546 7.8479 11.8823 0.8778 0.8689 0.0841 9.0087 13.98897 0.9311 0.9038 0.0203 6.4563 9.6542 0.9151 0.9092 0.0314 8.1132 11.66678 0.9875 0.9563 0.0116 4.3421 7.6542 0.9601 0.9587 0.0223 6.7681 8.932610 0.9227 0.9012 0.0478 7.7534 9.899 0.8831 0.8969 0.0573 9.2694 12.6432

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was set on 22 (see Table 5). The main reason behind thisdecline is that the space covered in the problem appears to betoo wide. The model efficiency was improved slightly whenthe number increased to 25. Therefore, it can be concluded thatthe optimum number of particles is 21. It should be also notedhere that as the number of particles increases, the greateramount of space is covered in the problem, leading to a sloweroptimization process.The same procedure was followed for other parameters of

the network, and the optimal configuration for the PSO-ANNnetwork was obtained as the following:

a) Number of hidden neurons = 8,b) Number of maximum iterations = 500,c) c1 and c2 = 2,d) Time interval = 0.0100, ande) Number of particles = 21.

Based on the above, the best ANN architecture is 3−8−1 (3input units, 8 neurons in the hidden layer, and 1 output neuron).A common question in neural networks is what is the sample

size of data required to train the network? No simpleprocedures have been proposed to answer this question.Certainly, it requires adequately large and representative datafor the neural network to learn. However, the number ofsamples for training depends on several parameters such asfactor selection and the structure of the neural networkmodel.88−91

Once the factor selection and neural network architecturewere determined, different sampling sizes were employed totrain while the testing phase of the PSO-ANN was being con-ducted using 45 real data of the South Pars gas field. The size oftraining data was selected between 50 and 500 data. Table 6shows the results of the network performance based on thestatistical investigations. Only performance of the training pro-cess in terms of MSE is shown in Figure 5. According to Table 6and Figure 5, it can be concluded that a sample of 250 data issufficient for training the PSO-ANN system. Therefore, 250 datapoints were chosen for the purpose of network training in thisstudy. The remaining 45 samples were put aside to be employedfor testing and validating the network’s integrity and robustness.In order to evaluate the performance of the PSO-ANN

algorithm, a back-propagation neural network (BP-ANN) wasapplied with the same data sets used in the PSO-ANN model.Predicted and measured normalized CGR values at the trainingand validation phases for the both BP-ANN and PSO-ANNmodels were compared as shown in Figures 6 and 7.The input variables of an ANN are frequently of various

types with different orders of magnitudes, such as temperature(T) and dewpoint pressure (Pd) in the case studied here. Thesame thing usually occurs for the output parameters. Thus, it isnecessary to normalize the input and output parameters basedon the ranges of data as they fall within a particular range. Forinstance, setting minimum −1 and the maximum +1 to have allthe data within the interval [−1, 1]. It should be mentionedhere that the legend of the vertical axis in Figures 6 and 7

Table 4. Performance of the PSO-ANN Model for Various Magnitudes of c1 and c2

training testing

c1 and c2 values R2 E2 MSE MEAE (%) MAAE (%) R2 E2 MSE MEAE (%) MAAE (%)

1.4 0.9344 0.8894 0.0784 8.9053 12.8762 0.8254 0.7945 0.0875 9.8144 13.89561.6 0.9498 0.9126 0.0679 7.0542 11.3453 0.8409 0.8131 0.0764 8.1635 12.43641.8 0.9559 0.9304 0.0489 6.1123 10.6402 0.8723 0.8389 0.0583 7.1234 11.57632.0 0.9789 0.9581 0.0206 4.7659 8.6574 0.9087 0.8563 0.0324 5.8548 8.74232.2 0.9673 0.9402 0.0341 5.9978 9.9897 0.8730 0.8379 0.0461 6.8863 10.44212.4 0.9514 0.9375 0.0452 7.4321 10.9769 0.8541 0.8231 0.0584 8.5627 11.7676

Table 5. Effect of Number of Particles on Performance of the PSO-ANN

training testing

number of particles R2 E2 MSE MEAE (%) MAAE (%) R2 E2 MSE MEAE (%) MAAE (%)

15 0.9694 0.9678 0.0875 7.4468 12.7845 0.8965 0.8911 0.0858 9.5476 14.875418 0.9367 0.9311 0.0903 9.1563 13.5473 0.8634 0.8549 0.0917 11.0765 15.435320 0.9711 0.9697 0.0869 6.5422 10.9863 0.9249 0.9167 0.0826 9.2254 12.986321 0.9897 0.9789 0.0764 4.9977 8.6451 0.9354 0.9231 0.0787 6.1066 9.123422 0.9706 0.9682 0.0964 6.8863 9.4821 0.9146 0.9078 0.0955 8.7789 11.546225 0.9719 0.9696 0.0947 6.7791 9.3789 0.9156 0.9110 0.0949 8.5673 11.2751

Table 6. Effect of Sampling Size on Performance of the Neural Network Model

training testing

number of training samples R2 E2 MSE MEAE (%) MAAE (%) R2 E2 MSE (%) MEAE MAAE (%)

50 0.9213 0.9189 0.075 9.6045 12.8256 0.9112 0.9098 0.0787 10.1807 13.9879100 0.9304 0.9279 0.065 8.4235 10.6446 0.9201 0.9187 0.0683 8.9289 11.6002150 0.9486 0.9382 0.055 7.2243 9.1455 0.9383 0.92883 0.0578 7.6577 9.9607200 0.9565 0.9496 0.048 6.3345 8.4261 0.9461 0.9402 0.0504 6.7145 9.2004250 0.9633 0.9571 0.044 5.4366 7.3422 0.9628 0.9475 0.0462 5.7627 8.1560300 0.9649 0.9581 0.043 5.3879 7.5611 0.9644 0.9485 0.0451 5.7111 8.2683400 0.9654 0.9588 0.042 5.4107 7.6423 0.9649 0.9492 0.0441 5.7353 8.3576500 0.9641 0.9576 0.042 5.4482 7.6988 0.9636 0.9480 0.0449 5.7750 8.4105

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presents normalized CGR that was calculated using thefollowing expression

=−−

−Normalized CGRCGR CGR

CGR CGR2( )

( )1min

max min (5)

where CGRmin and CGRmax are the minimum and maximumCGR of the data used in this study, respectively.PSO-ANN was trained by 50 generations followed by a BP

training procedure. The values of learning coefficient 0.7 andmomentum correction factor 0.001 were used for the back-propagation training algorithm.

As seen in Figure 7, there is a good agreement between themodel outputs and the experimental CGR, based on the testingdata simulated at this phase of the study. The simulationperformance of the PSO-ANN model was evaluated on thebasis of mean squared error (MSE), mean absolute error(MEAE), maximum absolute error (MAAE), and efficiencycoefficient (R2). The parameters MSE = 0.0617 and R2 = 0.9721for PSO-ANN in contrast with theMSE = 0.0981 and R2 = 0.8986for BP-ANN reveals that the PSO-ANN exhibits a betterperformance in prediction of CGR than BP-ANN (Figures 8and 9). It should be noted here that an R2 value greater than0.9 generally indicates a very satisfactory model performance;while an R2 magnitude in the range of 0.8−0.9 presents a goodperformance and a value less than 0.8 shows an unsatisfactorymodel performance, based on the statistical analysis.94,95

Figure 5. Training performance (MSE) vs sampling size.

Figure 6. Measured vs predicted CGR using the BP-ANN model: a)training and b) testing.

Figure 7. Measured vs predicted CGR using the PSO-ANN model: a)training and b) testing.

Table 7. Performances of the PSO-ANN Model, the BP-ANN Model, and a Correlation Proposed by Fattah et al.(2009)38

statistical parameter PSO-ANN BP-ANN Fattah et al. correlation

MSE 0.0617 0.0981 0.1054MEAE (%) 2.4503 6.7864 7.6654MAAE (%) 6.9114 10.6875 11.7112R2 0.9721 0.8986 0.8579E2 0.9497 0.8878 0.8432

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The extent of the match between the measured andpredicted CGR values by BP-ANN and PSO-ANN networksin term of a scatter diagram are shown in Figures 8 and 9.

These results convey the message that PSO-ANN has thecapability to avoid being trapped in local minima. Thisadvantage is due to the combination of global searching abilityof PSO and local searching ability of BP.The performance plots of the BP-ANN and developed PSO-

ANN models are depicted in Figures 10 and 11, respectively. Thecurves included in the figures present the relationship between thetesting, training, validation, and best models introduced for pre-dicting CGR in terms of MSE versus number of epochs. As shownwith green circles on Figures 10 and 11, the best performance forthe validation phase is 0.0022 at epoch about 3 for the BP-ANNmodel, while the PSO-ANN model experiences the best per-formance (MSE ≈ 0.0015) for the validation phase at epoch of 13.A comparison between CGR values obtained from a correlation

introduced by Fattah et al. (2009)38 and the proposed PSO-ANNand BP-ANNmodels is presented here to examine the performanceof the PSO-ANN model. Table 7 shows the MSE, MEAE, MAAE,E2, and R2 values for these different approaches in the validationphase. It can be concluded that the performance of PSO-ANN isbetter than the BP-ANN model and also the empirical correlation.An ANN is a learning structure in which the network is

trained by information of a certain process or system. In general,this training stage needs considerably more processing time thanwhat is required to evaluate the system using a single simulation

run or an empirical equation. In Table 8, the training time isrecorded for both the 39-bus system and the 1844-bus system,employed for the neural networks. For both systems, 250sample data were used for training the BP-ANN and PSO-ANN.The computations were conducted using a HP laptop, IntelPentium Dual CPU E2200 @ 2.8 GHz. Table 8 indicates that thepractical system (1844-bus) needs higher training time than the39-bus system as more input data were considered in the practical

Figure 8. Performance of the developed BP-ANN model based on R2:a) training and b) testing.

Figure 9. Performance of the developed PSO-ANN model based onR2: a) training and b) testing.

Figure 10. Performance plot of the proposed BP-ANN model in termsof number of epochs.

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system. Also, the table shows that the training process for the BP-ANN is faster than that for the PSO-ANN model.The main benefit of applying computational intelligence

technology to substitute the conventional experimental andnumerical methods for calculation of CGR is its possibility ofreal-time operation. ANN has the ability to satisfactorily shrinkwhat it learns in training process and broaden this to generateacceptable outputs for those inputs not dealt over the trainingstage. Once trained, the ANN system can forecast the value ofCGR for a given set of input data very fast as the computationin an ANN model does not involve any iteration which is thecase in numerical methods. Table 9 shows the comparison of

the computational speed of the BP-ANN and PSO-ANN. Basedon the results presented in the table, using the BP-ANN toestimate CGR is a little faster than the method; however, thePSO-ANN model exhibits a higher accuracy in estimating CGR.A sensitivity analysis was conducted for the newly developed

model using the analysis of variance (ANOVA) technique.94,95

The dependence of the dependent variable (CGR) on each ofthe independent variables was examined using the ANOVAtechnique. The results of the sensitivity analysis are presentedin Figure 12. The higher correlation between any input variableand the output variable indicates greater significance of thevariable on the magnitude of the dependent parameter. Clearly,

the mixture molecular weight is the most important parameteraffecting the CGR. An increase in the molecular weight leads toan increase in the CGR value. This finding is in agreement withseveral previously published research.33−38,41−85 In addition,impact of dewpoint pressure on the CGR supports the validity ofthe empirical correlations available in the literature.33−38,41−85

Increase in dewpoint pressure results in an increase in the valueof CGR. It should be also noted here that as the reservoir tem-perature increases, the magnitude of the CGR increases.The CGR for a particular gas sample of South Pars field was

obtained 31.8 and 31.5 STB/MMSCF based on the laboratorywork and the developed PSO-ANN model, respectively. Thesmall difference between the experimental and modeling resultsindicates the capability and high accuracy of the proposedPSO-ANN model in prediction of the CGR.

8. CONCLUSIONSAn artificial neural network intelligent system optimized withparticle swarm optimization was developed and tested using theexperimental data from South Pars gas field and the literature topredict condensate-to-gas ratio (CGR).Based on the results of this study, the following main

conclusions can be drawn:

1- For a particular sample of South Pars field, the CGR valuesobtained from the experimental work and PSO-ANN are31.8 and 31.5 STB/MMSCF, respectively. An acceptableagreement between these two determination approaches wasobserved.

2- Prediction of CGR was made possible using PSO-ANN,BP-ANN, and an empirical correlation. Comparing theresults of this study indicates that predictive performance ofthe proposed PSO-ANN is better than the BP-ANN modeland also the empirical correlation used in this paper.

3- For predicting the CGR, the PSO-ANN model has thecapability to avoid being trapped in local optimums dueto the combination of global searching ability of PSO andlocal searching ability of BP.

4- Finding an optimum structure for the PSO-ANN modelis an important feature in CGR prediction. The optimumconfiguration for the proposed model includes 8 hiddenneurons and the acceleration constants (c1 and c2) equalto 2 determined by using a trial and error technique.

5- A sensitivity analysis using the ANOVA techniqueshowed that molecular weight of mixture has the most

Figure 11. Performance plot of the proposed PSO-ANN model interms of number of epochs.

Table 8. CPU Time for the Neural Networks Employed inthe Study

type of neural network type of system training CPU time (s)

BP-ANN 39-bus system 671844-bus system 1924

PSO-ANN 39-bus system 1391844-bus system 3895

Table 9. Comparison of the Computation Time Cost for theBP-ANN and PSO-ANN Techniques

type of neural network type of system CPU time (s)

BP-ANN 39-bus system 0.051844-bus system 0.26

PSO-ANN 39-bus system 0.081844-bus system 0.32

Figure 12. Relative effects of important independent variables on CGR.

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effect on the magnitude of the CGR predicted. Also, asthe temperature increases, the CGR value increases.

6- Although the proposed ANN-PSO model works well forCGR prediction, it suffers from early convergence by thedegeneracy of several dimensions; even if no local optimaexist for cases considered with this model. Hence, abetter evolutionary algorithm is required to be combinedwith PSO and ANN which is a part of our future study.

■ APPENDIX AA part of experimental data employed in this study to develop thePSO-ANN model is listed in Table A1. To build the trainingnetwork, a large volume of the CGR data available in the literaturewas used to obtain an appropriate neural network model.

■ APPENDIX BR2, E2, MSE, MEAE, and MAAE are the statistical parametersto test the accuracy of the PSO-ANN model compared withcommon correlations. The equations to compute the aboveparameters and also the corresponding description are as follows:R2 is a statistic parameter which gives information on

goodness of fit in a model. In regression analysis or fitting of amodel, the R2 coefficient is a statistical measure of how well themodel line approximates the real data points. An R2 of 1.0indicates the model line perfectly fits the data. The followingequation shows the mathematical definition of R2:94,95

∑

∑=

−

−

=

=

R

CGR CGR

CGR CGR

( )

( )

2 i

n

iP M

i

n

iM M

1

2

1

2

(B1)

where CGRM and CGRP are the measured CGR and predictedCGR, respectively. CGRM represents the average of themeasured CGR data.The Nash−Sutcliffe model efficiency coefficient (E2) can be

employed to evaluate the predictive power of statistical andintelligent models. It is given as94,95

∑

∑= −

−

−

=

=

E

CGR CGR

CGR CGR1

( )

( )

2 i

n

iM

iP

i

n

iM M

1

2

1

2

(B2)

The mean squared error (MSE) is a measure of how close afitted line or developed model is to data points. The smaller themean squared error the closer the fit (or model) is to the actualdata. TheMSE has the units squared of whatever is plotted on thevertical axis. MSE is described by the following relationship94,95

∑=

−=MSE

CGR CGR

n

( )i

n

iM

iP

1

2

(B3)

where n is the number of samples.The other two performance measures that were used in this

paper to assess the effectiveness of the training and testing datainclude mean absolute error (MEAE) and maximum absoluteerror (MAAE) as follows94,95

∑=| − |

×=

MEAEn

CGR CGRCGR

%1

100i

niP

iM

iM

1 (B4)=

| − |×MAAE

CGR CGRCGR

%max

100iP

iM

iM

(B5)

Table A1. Experimental Data for CGR from VariousResearch Studies

run no. T (F) Pd (psi) Mw CGR (STB/MMSCF)

1 173 5871 18.13 19.72 253 4524 17.44 8.23 215 5393 21.27 32.54 261 4345 23.52 24.85 210 5570 25.23 7.66 203 4154 24.80 33.67 281 5228 23.98 5.68 250 5962 21.46 5.89 214 4564 23.94 36.110 177 4276 26.56 26.511 252 3589 21.81 16.612 240 3948 19.70 20.613 174 3292 24.79 28.914 179 4108 18.98 6.115 281 3968 18.96 27.716 224 4005 21.38 26.917 168 5471 23.00 19.218 171 4590 22.96 23.119 182 4558 17.79 15.420 270 5339 17.80 10.021 220 3321 19.49 17.322 231 4880 24.23 33.923 223 4019 20.82 11.724 173 3982 25.20 16.125 271 4723 26.32 14.926 199 5615 20.69 7.227 197 3563 17.05 20.928 235 5025 22.50 31.229 221 5732 24.35 33.430 208 3564 19.86 27.731 242 3870 21.68 16.832 276 4482 18.83 14.033 238 5695 26.36 26.634 198 4497 21.56 32.735 189 3805 22.17 34.136 217 4597 18.82 7.137 168 4722 17.51 19.938 268 4224 17.13 8.639 237 2996 25.10 12.640 189 5118 24.60 32.841 226 4513 18.55 10.842 171 4126 25.90 8.443 267 3150 23.69 21.944 217 4056 23.40 14.645 230 3673 26.90 16.746 232 3578 16.85 29.847 245 5461 20.95 31.448 209 4164 21.64 26.349 174 3115 22.63 17.850 219 4107 17.37 25.151 171 5341 23.43 11.452 252 3460 17.05 24.353 169 5764 22.35 26.154 278 3935 23.98 23.755 253 3967 21.62 15.6

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■ AUTHOR INFORMATIONCorresponding Author*Phone: 519-888-4567 ext. 36157. E-mail: szendehb@uwaterloo.ca. Corresponding author address: ChemicalEngineering Department, University of Waterloo, Waterloo,Ontario, Canada, N2L 3G1.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors would like to thank Ali Shafiei, University ofWaterloo, for his help and detailed technical attention in editingthe manuscript.

■ NOMENCLATUREAcronymsAI = artificial intelligenceANN = artificial neural networkBP = back propagationCCE = constant composition expansionCGR = condensate-to-gas ratioCVD = constant volume depletionEOS = equation of stateFL = fuzzy logicGA = genetic algorithmMAAE = maximum absolute errorMEAE = mean absolute errorMSE = mean squared errorMw = molecular weightNN = neural networkPSO = particle swarm optimizationP-T = pressure-temperaturePVT = pressure-volume-temperature

VariablesTj(k) = actual outputPgt = best ever particle position of particle i

Yj(k) = expected outputPit = global best position in the swarm until iteration t

Vit = velocity vector at iteration t

c1, c2 = trust parametersCi = hydrocarbon component (e.g., i = 1 refers to methane)E2 = Nash-Sutcliffe coefficientG = number of training samplesk = permeability (mD)m = number of output nodesn = number of samplesMw = molecular weightPd = dewpoint Pressurer1 , r2 = random numberR2 = coefficient of determinationT = reservoir temperatureXi = position of the i-th particlezi = overall composition of component i

Greek Letterω = inertia weightϕ = porosity

Subscriptsd = dewpointi = particle imax = maximummin = minimum

SuperscriptsM = measurednet = networkP = predicted

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