Research Article Developing a Robust Surrogate Model of ...downloads.hindawi.com/journals/mpe/2015/706897.pdf · 3. Artificial Neural Network and Particle Swarm Optimization Arti

Research ArticleDeveloping a Robust Surrogate Model of ChemicalFlooding Based on the Artificial Neural Network for EnhancedOil Recovery Implications

Mohammad Ali Ahmadi

Department of Petroleum Engineering Ahwaz Faculty of Petroleum Engineering Petroleum University of TechnologyPO Box 63431 Ahwaz 6198144471 Iran

Correspondence should be addressed to Mohammad Ali Ahmadi ahmadi6776yahoocom

Received 10 June 2014 Revised 5 October 2014 Accepted 13 October 2014

Academic Editor Shifei Ding

Copyright copy 2015 Mohammad Ali Ahmadi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Application of chemical flooding in petroleum reservoirs turns into hot topic of the recent researches Development strategies of theaforementioned technique are more robust and precise when we consider both economical points of view (net present value NPV)and technical points of view (recovery factor RF) In current study many attempts have been made to propose predictive modelfor estimation of efficiency of chemical flooding in oil reservoirs To gain this end a couple of swarm intelligence and artificialneural network (ANN) is employed Also lucrative and high precise chemical flooding data banks reported in previous attentionsare utilized to test and validate proposed intelligent model According to the mean square error (MSE) correlation coefficient andaverage absolute relative deviation the suggested swarm approach has acceptable reliability integrity and robustness Thus theproposed intelligent model can be considered as an alternative model to predict the efficiency of chemical flooding in oil reservoirwhen the required experimental data are not available or accessible

1 Introduction

The oil and gas upstream industries are recently encoun-tered with the difficulties and challenges of dealing withhydrocarbon resources whose productionswith conventionaltechnologies are following an upward trend of technicallimitations It is because of achieving the stage of declinephase by most of oilfields around the world Thereforehow to postpone the abandonment of reservoirs has tunedinto the priority of researchers in the worldwide Theirresearches normally highlight the concept of great necessitiesfor inventions of new techniques normally classified astertiary oil recovery methods having abilities of maintainingthe economic production rate [1ndash3]

Chemical enhanced oil recovery approaches as one ofthe most effective subsets of tertiary methods are knownas a key to unlock the exploitation of referred resourcesDifferent methods for this process have been developedsuch as polymer surfactantpolymer (SP) and alkalinesur-factantpolymer (ASP) flooding These methods are applied

to increase the rate of oil production through focusing onboth lowering the interfacial tension and reducing the watermobility In more details it has enormously been declaredin previous literatures that in order to design manageand run a chemical enhanced oil recovery operation it ishighly required to set very expensive and time-consumingbut precise experimental procedures which their generatedresults must be gained to plan effectively the process ofinjecting chemical materials [4ndash9]

The laboratorial generated outputs are then used to con-clude two parameters recovery factor (RF) and net presentvalue (NPV) which are used to evaluate the performanceof the chemical flooding which is one of the most popularmethods of chemical enhanced oil recovery Having knowl-edge about these two parameters is essentially vital to makedecisions if it is beneficial to run the referred operationUnfortunately there are no global methods to interpretsimultaneously both aforementioned factors although thereare numerous numbers of different software and numerical or

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 706897 9 pageshttpdxdoiorg1011552015706897

2 Mathematical Problems in Engineering

Table 1 Statistical analysis of the implemented chemical flooding data samples [9]

Parameter Unit Type Min Max Average Standard deviationSurfactant slug size PV Input 0097 0259 0177 0072Surfactant concentration Vol fraction Input 0005 003 0017 0011Polymer concentration in surfactant slug wt Input 01 025 0177 0067Polymer drive size PV Input 0324 0648 0482 0144Polymer concentration in polymer drive wt Input 01 02 0148 0044119870V119870ℎ ratio mdash Input 001 025 0129 0107Salinity of polymer drive MeqmL Input 03 04 0349 0045Recovery factor (RF) Output 1482 5699 3967 924Net present value (NPV) $ MM Output 1781 7229 445 153

analytical methods which are capable of making very precisequantitative decisions about the amount of one of the RF orNPV [10ndash12]

Hence there is a great need in oilfield for having accessto a solution or model which can predict the amount ofthese two parameters at the same time The major aim ofcurrent study is to execute new kind of artificial intelligenceapproaches to suggest robust and accurate predictive methodto forecast efficiency of the chemical flooding throughpetroleum reservoirs To gain successfully this referred goalhybridization of artificial neural network and particle swarmoptimization (PSO) was executed on the previous literaturedata bases The integrity and performance of the proposedpredictive approaches in estimating recovery factor (RF) andnet present value (NPV) from the literature are described indetails

2 Data Gathering

Thedata utilized throughout this research have been gatheredfrom previous attentions [9] in which chemical floodinghad been simulated in Benoist sand reservoir by executingUTCHEM simulator That reservoir has been producedunder primary and secondary processes over fifty years Theoriginal dataset contained 202 data Each data had 7 inputssurfactant slug size surfactant concentration in surfactantslug polymer concentration in surfactant slug polymer drivesize and polymer concentration in polymer drive 119870V119870ℎratio and salinity of polymer drive In addition the outputswere RF and NPVThe ranges of implemented data banks arereported in Table 1 [9]

3 Artificial Neural Network and ParticleSwarm Optimization

Artificial neural network (ANN) includes simple nodesnamed as neurons which are bonded to each other toconstruct a network model Indeed the biological nervoussystems can be simulated with the ANN system somehowCharacterization of an ANN model is normally performedthrough three ways including (a) certain patterns betweenvarious layers (b) connection between input and output viaactivation function and (c) updating the interconnectionweights through training process [13ndash24]

In fact the main purpose of an ANN model is todetermine target function through internal computationduring the training phase if the values of input variables areprovided The most common type of ANN is the multilayerfeed forward neural network which is made up of groupof interconnected neurons organized in the form of layersinput layer hidden layer(s) and output layer where each layercomprises a group of neurons as presented in Figure 1 Thisnetwork is strictly an acyclic type since signals propagate onlyin a forward direction from the input neurons to the outputneurons and no signals are allowed to be fed-back amongthe neurons The number of neurons in the input and outputlayers is decided by the number of input and output variablesthat are planned for the predictive tool However the optimalnumber of neurons in hidden layer(s) is a strong function ofnonlinearity and dimensionality of the problem under study[13ndash24]

The artificial neuron is the fundamental part of the neuralnetworks Each artificial neuronmdashexcluding neurons at theinput layermdashtakes and processes inputs gathered from otherneurons Given further information each artificial neuron isa mathematical information-processing unit The processedinformation is presented at the output end of the neuronFigure 2 addresses the procedure inwhich an artificial neurontreats the data and information entered in the model Eachinput signal (119886119896) is primarily multiplied by the correspondingweight value (119908119896119895) and the resultant products are summedup to generate a total weight in the form of 11990811989511198861 + 11990811989521198862 +sdot sdot sdot + 119908119895119898119886119898 The sum of the weighted inputs and the bias(119878119895 = sum

119898

119896=1119908119895119896 sdot 119886119896 + 119887119895) forms the input to the activation

function 120593 An activation function processes this sum andgives out the output 119900119895 Indeed the resulting sum is processedby a neuron activation function to obtain the ultimate outputof the neuron as follows [13ndash26]

119900119895 = 120593 (119878119895) = 120593(

119898

sum

119896=1

119908119895119896 sdot 119886119896 + 119887119895) (1)

This output will be the input signal for the neurons in the fol-lowing layer The linear (purelin) transfer tan-sigmoid (tan-sig) activation and log-sigmoid (logsig) activation functionsare mostly employed in the practical cases with applicationsin science and engineering disciplines The corresponding

Mathematical Problems in Engineering 3

X1

X2

X3

X4

Y1

Y2

BiasBiasbH1

bO1wH

11

wH44

wO11

wO24

+1+1

Input layer Hidden layer Output layer

Figure 1 Architecture of multilayer feed forward neural network The symbol 119908119867119902119903 denotes the synaptic weight between the output of the119903th neuron in the hidden layer and the input of the 119902th neuron in output layer The symbol 119887119867

119902denotes the bias of the 119902th neuron in hidden

layer The superscript 119874 stands for output layer

Artificial neuron

bj

a1

a2

am

wj1

wj2

wjm

(Summing function)

(Bias)

(Activation function)

(Output)

Synaptic weights

Sj = sum akwjk + bj

Oj = 120593 Sj( )

( 120593 Sj( ))

Figure 2 Information processing by an artificial neuron

relationships for these functions are defined respectively by(2)ndash(4) as given below [13ndash27]

120593 (119904) = 119904 (2)

120593 (119904) =119890119904minus 119890minus119904

119890119904 + 119890minus119904 (3)

120593 (119904) =1

1 + 119890minus119904 (4)

The weight factors are generally considered as the adaptiveparameters in the network to obtain the strength of theinput signals A bias is characterized with a weight whichis not responsible for connecting an input of two neuronsto an output A particular level of a neuron output signalis represented by a set of bias that does not depend on theinput signals The weight factors and biases are tuned duringthe course of training phase such that the network is able toforecast the accurate target parameter for a given set of inputsThere are a number of training algorithms with differentmethodologies in the context of intelligence system A varietyof optimization tools such as particle swarm optimization(PSO) [15 18 19] genetic algorithm (GA) [21] hybrid genetic

algorithm and particle swarm optimization (HGAPSO) [1316] unified particle swarm optimization (UPSO) [14] andimperialist competitive algorithm (ICA) [17 20 23] forweight training of neural networks have been used

Kennedy [27] introduced the PSO as a strong stochasticoptimization technique which simulates the social mannersof birds within a group based on population concept Itsearches for an optimum solution by iteratively updating aswarm of particles

Themodel originally includes a group of randomparticles(solutions) A random velocity is attributed to each candidateparticle which flies within the problem space The solutionsconsist of memory and try to attain the best position orandfitness This parameter is symbolized by ldquo119901bestrdquo that is linkedonly to a specific particle The model also retains the bestfitness known as ldquo119892bestrdquo which is found among the entiresolutions (particles) in the swarm The candidate particlethat obtains this fitness is the global best in the population[25ndash28] In the current study a particlersquos fitness is calculatedthrough determination of the network output for every pointin the training part and then computing the sum of squaresof the resultant errors (MSE) for performance evaluation


Start

Stop

Choose swarm of particles size

Initialize each particle with random position and velocityeach particlersquos position contains a series of ANN weights

Generate new swarmFind fitness (f) value for each

particle in the swarm

Update position of eachparticle usingequation (6)

Update velocityusing equation (5)

Next iteration CriterionNo

Yes

Evolved ANN weights

they are ready for ANN testing

Iff(x) is better than f(gbest)

Iff(x) is better than f(pbest)set current value as the new pbest

set current value as the new gbest

Figure 3 PSO-based algorithm flowchart in optimization of the weights of ANN

The basic PSO theory involves variation of each particlevelocity toward its 119901best and 119892best locations at each timeinterval The particlesrsquo new velocity and position are updatedaccording to the following equations [13ndash28]

V119894119899+1= 120596V119894119899+ 11988811199031

119899[119909119894119901best

119899minus 119909119894119899]

+ 11988821199032119899[119909119892best

119899minus 119909119894119899]

(5)

119909119894119899+1= 119909119894119899+ V119894119899+1 (6)

where V119894119899 and V119894

119899+1 are velocities of particle 119894 at iterations119899 and 119899 + 1 119909119894

119899 and 119909119894119899+1 are positions of particle 119894 at

iterations 119899 and 119899 + 1 120596 represents the inertia weight thatdirects the exploitation and exploration of the search spaceas it continuously updates velocity 1198881 and 1198882 are termedas cognition and social components respectively They areconsidered as the acceleration constants which alter thevelocity of a solution in the direction of119901best and119892best [13ndash28]and 1199031

119899 and 1199032119899 refer to the two random variables uniformly

distributed in the interval of [0 1]Herein PSO algorithm has been used in evolving weights

of multilayer feed forward neural network In this case aparticlersquos position at any iteration is described as a particlewhose coordinates are connection weights The vectors ofweights for each particle 119894 will be called 119909119894 Throughoutthe training process the above equations (equations (5) and(6)) will customize the network weights until a criterionis met In this case a lower MSE as a sufficiently good

fitness is achieved nevertheless a maximum number ofiterations are used to terminate the iterative search processif no improvement is observed over a number of consecutivegenerations in an appropriate timeThe flowchart of the PSO-based training algorithm for the ANN is shown in Figure 3

The PSO utilizes a random procedure in the search spaceof the problem such that particles in the population aredirected toward optimum positions but not in or betweenoptimal areas [27] Thus PSO can be used to train neuralnetworks with nondifferentiable (even discontinuous) neu-rons activation functions It can be also implemented in caseswhere gradient or error information is not accessible PSOis easy to implement and there are few parameters to beadjusted However the uniqueness of the algorithm lies in thedynamic interactions among the particles that turn it into asocial-psychological model of knowledge management [27]

4 Results and Discussion

According to the study accomplished by Cybenko [29] anetwork that consists of only one single hidden layer has theability to approximate nearly any kind of nonlinear functionHowever determination of the ideal number of neurons inthe hidden layer is a challenging task few neurons will notgive adequate precision and too many hidden neurons maylead to overfitting It means that the training data might befitted adequately however considerable oscillations betweenthe points are noticed in the fitting curve resulting in poor


084086088090920940960981

0

002

004

006

008

01

012

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(a)

086088090920940960981

0002004006008

01012014016

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(b)

Figure 4 Effect of number of hidden neuron on PSO-ANN accuracy of (a) recovery and (b) NPVpredictions in terms ofMSE and119877-squared

0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t

Actual recovery factor (RF)

DataFit R2 = 09994

(a)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09997

(b)

Figure 5 Performance plot of the suggested network model for determining recovery factor of chemical flooding owing to correlationcoefficient (1198772) (a) training phase and (b) testing phase

interpolation and extrapolation The network performanceis evaluated as demonstrated in Figure 4 when differentnumber of neurons is tested A smart model with one hiddenlayer (including just one neuron) was primarily built in thecurrent study to predict the recovery factor and net presentvalue (NPV) of chemical flooding in oil reservoirs Predictionaccuracy was further analyzed by an increase in the numberof neurons to 10 to decide on the most precise techniqueAs clear from the results demonstrated in Figure 4 a 3-6-1 architecture (6 neurons in the hidden layer 3 neuronsin the input layer and one neuron in output layer) offersthe best model for recovery factor and net present value(NPV) prediction in terms ofMSE and1198772 since the optimumstructure achieved by the trial and error procedure has a verylow mean squared error of MSE = 00012 and a satisfactorycoefficient of determination of 1198772 = 09996 on the basis ofcomparison between the predicted and real data

The generated results of the proposed intelligent approachare depicted through Figures 5 to 10 The existing contrastsbetween suggested intelligent approach and related recoveryfactor (RF) of the chemical flooding in oil reservoir in theregression plot have been depicted in Figure 5 As shownin Figure 5 which is a graphical and scatter presentation

of the PSO-ANN results versus corresponding determinedrecovery factor (RF) data the PSO-ANN outputs lie overthe line 119884 = 119883 the fact that indicates the identity ofoutputs gained from suggestedPSO-ANNmodel and relevantrecovery factor data samples To serve better understandingabout generated results of the proposed PSO-ANN modelthe comparison between gained recovery factor from theaddressedmodel and real recovery factor data versus relevantdata index has been illustrated in Figure 6 As illustrated inFigure 6 the obtained results of proposed model are as closeas possible to real recovery factor (RF) data samples To put itanother way the outputs of the PSO-ANN approach have thesame behaviour as actual data doThe high considerable levelof efficiency and accuracy related to the PSO-ANN approachin prediction of the recovery factor dataset of chemicalflooding has once again been certified in Figure 6 Moreoverthe robustness of the PSO-ANN has been demonstratedin terms of the relative deviations of PSO-ANN modeloutputs from corresponding determined recovery factor datain Figure 7 As could be observed in Figure 7 the highestdeviations of the suggested approach results are subjected tothe early boundary of recovery factor data samples 5 is themaximum degree of relative deviation shown in Figure 7


0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Actu

al re

cove

ry fa

ctor

(RF)

Data index

Experimental dataPSO-ANN output

(a)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

Actu

al re

cove

ry fa

ctor

(RF)

Data index

(b)

Figure 6 Comparison between suggested network model and recovery factor versus relevant data index (a) training phase and (b) testingphase

0

2

4

6

8

10

0 10 20 30 40 50 60

Dev

iatio

n (

)


Training dataTesting data

minus10

minus8

minus6

minus4

minus2

Figure 7 Relative error distribution of the proposed approach versus actual recovery factor (RF)

The draw parallel between our proposed intelligent PSO-ANN model results and related net present value (NPV) ofthe chemical flooding in oil reservoir in the regression plothas been shown in Figure 8 As shown in Figure 8 whichis a graphical and scatter presentation of the PSO-ANNresults versus corresponding determined net present value(NPV) data the PSO-ANN outputs lie over the line 119884 = 119883the fact that indicates the identity of outputs gained fromsuggested PSO-ANN model and relevant net present value(NPV) data samples The comparison between generated netpresent value (NPV) from the addressed approach and realnet present value (NPV) data versus relevant data indexhas been shown in Figure 9 As illustrated in Figure 9 theobtained results of proposed model are as close as possibleto net present value (NPV) data samples To put it anotherway the outputs of the PSO-ANN approach have the samebehaviour as actual data do Furthermore the effectivenessof the proposed intelligent model has been depicted interms of the relative deviations of PSO-ANN model outputs

Table 2 Statistical parameters of the proposed approaches in pre-diction of efficiency of chemical flooding in oil reservoirs

PSO-ANNRF NPV

Correlation coefficient (1198772) 09997 09996Mean square error (MSE) 00012 00015Mean absolute error (MAE) 0098 00206

from corresponding indicated net present value (NPV) datain Figure 10 As can be seen from Figure 10 the highestdeviations of the suggested approach results are subjected tothe early boundary of net present value (NPV) data 6 is themaximum degree of relative deviation depicted in Figure 10

The performance efficiency of the selected network isassessed using the various error analysis parameters Table 2tabulates the PSO-ANNaccuracy in terms of correlation coef-ficient (119877) coefficient of determination (1198772) mean absolute


012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)

Figure 8 Performance plot of the suggested network model for determining net present value (NPV) of chemical flooding owing to corre-lation coefficient (1198772) (a) training phase and (b) testing phase

012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)

Figure 9 Comparison between suggested network model and net present value (NPV) versus relevant data index (a) training phase and (b)testing phase

0 1 2 3 4 5 6 7 8Net present value


02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2

Figure 10 Relative error distribution of the proposed approach versus actual net present value (NPV)


error (MAE) and mean squared error (MSE) which aredefined as follows

119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)

in which 119873 represents the total number of data pointsincluding either training testing or whole data set (input andoutput pairs) 119910119879

119894refers to the actual value at the sampling

point 119894119910119875119894is the 119894th output of themodel and119910119879 and119910119875 stand

for the average magnitudes of the actual and predicted datarespectively

5 Conclusions

Owing to the gained results of this contribution the followingmajor conclusions can be drawn

(1) Adequate agreement between gain dew point pres-sure from the developed intelligent model and cor-responding real recovery factornet present value(NPV) values is observed In other words the con-ventional approaches fail to monitor real recoveryfactornet present value (NPV) of chemical floodingdedicated to the gained statistical criteria such asmean square error (MSE) and correlation coefficient

(2) The evolved intelligent network model for monitor-ing real recovery factornet present value (NPV) ofchemical flooding is user friendly fast and cheapfor implementation Moreover it is very useful anduser friendly for evolving the accuracy and robustnessof the commercial reservoir simulators like ECLIPSEand computer modelling group (CMG) software forenhanced oil recovery (EOR) from oil reservoirs

Nomenclature

Abbreviations

ANN Artificial neural networkBP Back propagationGA Genetic algorithmHGAPSO Hybrid genetic algorithm and particle swarm

optimizationICA Imperialist competitive algorithm

MAE Mean absolute error (MAE)MSE Mean squared error (MSE)PSO Particle swarm optimizationPSO Particle swarm optimization1198772 Coefficient of determination

UPSO Unified particle swarm optimization

Variables

119910119875 The average of the predicted data119910119879 The average of the actual data119887119895 Bias1198881 Cognition component1198882 Social components119873 The total number of data points119900119895 Output1199031119899 and 1199032

119899 Two random numbers119878119895 Sum of interconnection weightsV119894 Velocity of particle 119894119882119895119894 Interconnection weights in network model119909119894 Position of particle 119894119910119894119875 The 119894th output of the model

119910119894119879 The actual at the sampling point 119894

Greek Letters120575 Absolute relative error120593 Activation function120596 The inertia weight

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] B Shaker Shiran and A Skauge ldquoEnhanced oil recovery (EOR)by combined low salinity waterpolymer floodingrdquo Energy andFuels vol 27 no 3 pp 1223ndash1235 2013

[2] S Strand T Puntervold and T Austad ldquoEffect of temperatureon enhanced oil recovery frommixed-wet chalk cores by spon-taneous imbibition and forced displacement using seawaterrdquoEnergy and Fuels vol 22 no 5 pp 3222ndash3225 2008

[3] H ZhangMDong and S Zhao ldquoWhich one ismore importantin chemical flooding for enhanced court heavy oil recov-ery lowering interfacial tension or reducing water mobilityrdquoEnergy amp Fuels vol 24 no 3 pp 1829ndash1836 2010

[4] A Z Abidin T Puspasari and W A Nugroho ldquoPolymers forenhanced oil recovery technologyrdquo Procedia Chemistry vol 4pp 11ndash16 2012

[5] M A Ahmadi Y Arabsahebi S R Shadizadeh and S Shokrol-lahzadeh Behbahani ldquoPreliminary evaluation of mulberry leaf-derived surfactant on interfacial tension in an oil-aqueoussystem EOR applicationrdquo Fuel vol 117 pp 749ndash755 2014

[6] P Luo Y Zhang and S Huang ldquoA promising chemical-augmented WAG process for enhanced heavy oil recoveryrdquoFuel vol 104 pp 333ndash341 2013

[7] AMandal A Samanta A Bera andKOjha ldquoCharacterizationof oil-water emulsion and its use in enhanced oil recoveryrdquo


Industrial and Engineering Chemistry Research vol 49 no 24pp 12756ndash12761 2010

[8] A Sabhapondit A Borthakur and I Haque ldquoWater solubleacrylamidomethyl propane sulfonate (AMPS) copolymer as anenhanced oil recovery chemicalrdquo Energy and Fuels vol 17 no3 pp 683ndash688 2003

[9] M S Karambeigi R Zabihi andZHekmat ldquoNeuro-simulationmodeling of chemical floodingrdquo Journal of Petroleum Scienceand Engineering vol 78 no 2 pp 208ndash219 2011

[10] MAAhmadi and S R Shadizadeh ldquoImplementation of a high-performance surfactant for enhanced oil recovery from carbon-ate reservoirsrdquo Journal of Petroleum Science and Engineeringvol 110 pp 66ndash73 2013

[11] G Chen X Wang Z Liang et al ldquoTontiwachwuthikul Psimulation of CO2-oil Minimum Miscibility Pressure (MMP)for CO2 Enhanced Oil Recovery (EOR) using neural networksrdquoEnergy Procedia vol 37 pp 6877ndash6884 2013

[12] N Loahardjo X Xie and N R Morrow ldquoOil recovery bysequential waterflooding of mixed-wet sandstone and lime-stonerdquo Energy amp Fuels vol 24 no 9 pp 5073ndash5080 2010

[13] M Ali Ahmadi andMGolshadi ldquoNeural network based swarmconcept for prediction asphaltene precipitation due to naturaldepletionrdquo Journal of Petroleum Science and Engineering vol98-99 pp 40ndash49 2012

[14] M A Ahmadi ldquoNeural network based unified particle swarmoptimization for prediction of asphaltene precipitationrdquo FluidPhase Equilibria vol 314 pp 46ndash51 2012

[15] S Zendehboudi M A Ahmadi L James and I Chatzis ldquoPre-diction of condensate-to-gas ratio for retrograde gas condensatereservoirs using artificial neural network with particle swarmoptimizationrdquo Energy amp Fuels vol 26 no 6 pp 3432ndash34472012

[16] M Ali Ahmadi S Zendehboudi A Lohi A Elkamel and IChatzis ldquoReservoir permeability prediction by neural networkscombined with hybrid genetic algorithm and particle swarmoptimizationrdquo Geophysical Prospecting vol 61 no 3 pp 582ndash598 2013

[17] S Zendehboudi M A Ahmadi O Mohammadzadeh ABahadori and I Chatzis ldquoThermodynamic investigation ofasphaltene precipitation during primary oil production labora-tory and smart techniquerdquo Industrial andEngineeringChemistryResearch vol 52 no 17 pp 6009ndash6031 2013

[18] S Zendehboudi M A Ahmadi A Bahadori A Shafiei and TBabadagli ldquoA developed smart technique to predict minimummiscible pressuremdashEOR implicationsrdquo Canadian Journal ofChemical Engineering vol 91 no 7 pp 1325ndash1337 2013

[19] M A Ahmadi and S R Shadizadeh ldquoNew approach forprediction of asphaltene precipitation due to natural depletionby using evolutionary algorithm conceptrdquo Fuel vol 102 pp716ndash723 2012

[20] M A Ahmadi ldquoPrediction of asphaltene precipitation usingartificial neural network optimized by imperialist competitivealgorithmrdquo Journal of Petroleum Exploration and ProductionTechnology vol 1 no 2ndash4 pp 99ndash106 2011

[21] S Zendehboudi A R Rajabzadeh A Bahadori et al ldquoConnec-tionist model to estimate performance of steam-assisted gravitydrainage in fractured and unfractured petroleum reservoirsenhanced oil recovery implicationsrdquo Industrial and EngineeringChemistry Research vol 53 no 4 pp 1645ndash1662 2014

[22] M A Ahmadi M Ebadi and S M Hosseini ldquoPredictionbreakthrough time of water coning in the fractured reser-voirs by implementing low parameter support vector machineapproachrdquo Fuel vol 117 part A pp 579ndash589 2014

[23] M A Ahmadi M Ebadi A Shokrollahi and S M J MajidildquoEvolving artificial neural network and imperialist competitivealgorithm for prediction oil flow rate of the reservoirrdquo AppliedSoft Computing Journal vol 13 no 2 pp 1085ndash1098 2013

[24] M A Ahmadi and M Ebadi ldquoEvolving smart approach fordetermination dew point pressure through condensate gasreservoirsrdquo Fuel vol 117 part B pp 1074ndash1084 2014

[25] M A Ahmadi R Soleimani and A Bahadori ldquoA computa-tional intelligence scheme for prediction equilibriumwater dewpoint of natural gas in TEG dehydration systemsrdquo Fuel vol 137pp 145ndash154 2014

[26] M A Ahmadi M Ebadi and A Yazdanpanah ldquoRobustintelligent tool for estimating dew point pressure in retrogradedcondensate gas reservoirs application of particle swarm opti-mizationrdquo Journal of Petroleum Science and Engineering 2014

[27] J Kennedy ldquoParticle swarm social adaptation of knowledgerdquoin Proceedings of the IEEE International Conference on Evolu-tionaryComputation (ICEC rsquo97) pp 303ndash308 Indianapolis IndUSA April 1997

[28] M-A AhmadiMMasumi R Kharrat andAHMohammadildquoGas analysis by in situ combustion in heavy-oil recovery pro-cess experimental andmodeling studiesrdquoChemical Engineeringamp Technology vol 37 no 3 pp 409ndash418 2014

[29] G V Cybenko ldquoApproximation by superpositions of a sig-moidal functionrdquoMathematics of Control Signals and Systemsvol 2 no 4 pp 303ndash314 1989

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Statistical analysis of the implemented chemical flooding data samples [9]

Parameter Unit Type Min Max Average Standard deviationSurfactant slug size PV Input 0097 0259 0177 0072Surfactant concentration Vol fraction Input 0005 003 0017 0011Polymer concentration in surfactant slug wt Input 01 025 0177 0067Polymer drive size PV Input 0324 0648 0482 0144Polymer concentration in polymer drive wt Input 01 02 0148 0044119870V119870ℎ ratio mdash Input 001 025 0129 0107Salinity of polymer drive MeqmL Input 03 04 0349 0045Recovery factor (RF) Output 1482 5699 3967 924Net present value (NPV) $ MM Output 1781 7229 445 153

analytical methods which are capable of making very precisequantitative decisions about the amount of one of the RF orNPV [10ndash12]

Hence there is a great need in oilfield for having accessto a solution or model which can predict the amount ofthese two parameters at the same time The major aim ofcurrent study is to execute new kind of artificial intelligenceapproaches to suggest robust and accurate predictive methodto forecast efficiency of the chemical flooding throughpetroleum reservoirs To gain successfully this referred goalhybridization of artificial neural network and particle swarmoptimization (PSO) was executed on the previous literaturedata bases The integrity and performance of the proposedpredictive approaches in estimating recovery factor (RF) andnet present value (NPV) from the literature are described indetails

2 Data Gathering

Thedata utilized throughout this research have been gatheredfrom previous attentions [9] in which chemical floodinghad been simulated in Benoist sand reservoir by executingUTCHEM simulator That reservoir has been producedunder primary and secondary processes over fifty years Theoriginal dataset contained 202 data Each data had 7 inputssurfactant slug size surfactant concentration in surfactantslug polymer concentration in surfactant slug polymer drivesize and polymer concentration in polymer drive 119870V119870ℎratio and salinity of polymer drive In addition the outputswere RF and NPVThe ranges of implemented data banks arereported in Table 1 [9]

3 Artificial Neural Network and ParticleSwarm Optimization

Artificial neural network (ANN) includes simple nodesnamed as neurons which are bonded to each other toconstruct a network model Indeed the biological nervoussystems can be simulated with the ANN system somehowCharacterization of an ANN model is normally performedthrough three ways including (a) certain patterns betweenvarious layers (b) connection between input and output viaactivation function and (c) updating the interconnectionweights through training process [13ndash24]

In fact the main purpose of an ANN model is todetermine target function through internal computationduring the training phase if the values of input variables areprovided The most common type of ANN is the multilayerfeed forward neural network which is made up of groupof interconnected neurons organized in the form of layersinput layer hidden layer(s) and output layer where each layercomprises a group of neurons as presented in Figure 1 Thisnetwork is strictly an acyclic type since signals propagate onlyin a forward direction from the input neurons to the outputneurons and no signals are allowed to be fed-back amongthe neurons The number of neurons in the input and outputlayers is decided by the number of input and output variablesthat are planned for the predictive tool However the optimalnumber of neurons in hidden layer(s) is a strong function ofnonlinearity and dimensionality of the problem under study[13ndash24]

The artificial neuron is the fundamental part of the neuralnetworks Each artificial neuronmdashexcluding neurons at theinput layermdashtakes and processes inputs gathered from otherneurons Given further information each artificial neuron isa mathematical information-processing unit The processedinformation is presented at the output end of the neuronFigure 2 addresses the procedure inwhich an artificial neurontreats the data and information entered in the model Eachinput signal (119886119896) is primarily multiplied by the correspondingweight value (119908119896119895) and the resultant products are summedup to generate a total weight in the form of 11990811989511198861 + 11990811989521198862 +sdot sdot sdot + 119908119895119898119886119898 The sum of the weighted inputs and the bias(119878119895 = sum

119898

119896=1119908119895119896 sdot 119886119896 + 119887119895) forms the input to the activation

function 120593 An activation function processes this sum andgives out the output 119900119895 Indeed the resulting sum is processedby a neuron activation function to obtain the ultimate outputof the neuron as follows [13ndash26]

119900119895 = 120593 (119878119895) = 120593(

119898

sum

119896=1

119908119895119896 sdot 119886119896 + 119887119895) (1)

This output will be the input signal for the neurons in the fol-lowing layer The linear (purelin) transfer tan-sigmoid (tan-sig) activation and log-sigmoid (logsig) activation functionsare mostly employed in the practical cases with applicationsin science and engineering disciplines The corresponding


X1

X2

X3

X4

Y1

Y2

BiasBiasbH1

bO1wH

11

wH44

wO11

wO24

+1+1





Artificial neuron

bj

a1

a2

am

wj1

wj2

wjm

(Summing function)

(Bias)


(Output)

Synaptic weights

Sj = sum akwjk + bj

Oj = 120593 Sj( )

( 120593 Sj( ))



120593 (119904) = 119904 (2)

120593 (119904) =119890119904minus 119890minus119904

119890119904 + 119890minus119904 (3)

120593 (119904) =1

1 + 119890minus119904 (4)






Start

Stop








Yes

Evolved ANN weights







V119894119899+1= 120596V119894119899+ 11988811199031

119899[119909119894119901best

119899minus 119909119894119899]

+ 11988821199032119899[119909119892best

119899minus 119909119894119899]

(5)

119909119894119899+1= 119909119894119899+ V119894119899+1 (6)

where V119894119899 and V119894












084086088090920940960981

0

002

004

006

008

01

012

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(a)

086088090920940960981

0002004006008

01012014016

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(b)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09994

(a)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09997

(b)






0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Actu

al re

cove

ry fa

ctor

(RF)

Data index


(a)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

Actu

al re

cove

ry fa

ctor

(RF)

Data index

(b)


0

2

4

6

8

10

0 10 20 30 40 50 60

Dev

iatio

n (

)



minus10

minus8

minus6

minus4

minus2




PSO-ANNRF NPV





012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)


012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)




02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2




119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X1

X2

X3

X4

Y1

Y2

BiasBiasbH1

bO1wH

11

wH44

wO11

wO24

+1+1





Artificial neuron

bj

a1

a2

am

wj1

wj2

wjm

(Summing function)

(Bias)


(Output)

Synaptic weights

Sj = sum akwjk + bj

Oj = 120593 Sj( )

( 120593 Sj( ))



120593 (119904) = 119904 (2)

120593 (119904) =119890119904minus 119890minus119904

119890119904 + 119890minus119904 (3)

120593 (119904) =1

1 + 119890minus119904 (4)






Start

Stop








Yes

Evolved ANN weights







V119894119899+1= 120596V119894119899+ 11988811199031

119899[119909119894119901best

119899minus 119909119894119899]

+ 11988821199032119899[119909119892best

119899minus 119909119894119899]

(5)

119909119894119899+1= 119909119894119899+ V119894119899+1 (6)

where V119894119899 and V119894












084086088090920940960981

0

002

004

006

008

01

012

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(a)

086088090920940960981

0002004006008

01012014016

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(b)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09994

(a)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09997

(b)






0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Actu

al re

cove

ry fa

ctor

(RF)

Data index


(a)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

Actu

al re

cove

ry fa

ctor

(RF)

Data index

(b)


0

2

4

6

8

10

0 10 20 30 40 50 60

Dev

iatio

n (

)



minus10

minus8

minus6

minus4

minus2




PSO-ANNRF NPV





012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)


012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)




02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2




119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

Stop








Yes

Evolved ANN weights







V119894119899+1= 120596V119894119899+ 11988811199031

119899[119909119894119901best

119899minus 119909119894119899]

+ 11988821199032119899[119909119892best

119899minus 119909119894119899]

(5)

119909119894119899+1= 119909119894119899+ V119894119899+1 (6)

where V119894119899 and V119894












084086088090920940960981

0

002

004

006

008

01

012

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(a)

086088090920940960981

0002004006008

01012014016

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(b)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09994

(a)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09997

(b)






0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Actu

al re

cove

ry fa

ctor

(RF)

Data index


(a)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

Actu

al re

cove

ry fa

ctor

(RF)

Data index

(b)


0

2

4

6

8

10

0 10 20 30 40 50 60

Dev

iatio

n (

)



minus10

minus8

minus6

minus4

minus2




PSO-ANNRF NPV





012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)


012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)




02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2




119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










084086088090920940960981

0

002

004

006

008

01

012

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(a)

086088090920940960981

0002004006008

01012014016

0 2 4 6 8 10 12

Cor

relat

ion

coeffi

cien

t

Mea

n sq

uare

erro

r (M

SE)

Number of neurons

(b)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09994

(a)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

PSO

-AN

N o

utpu

t


DataFit R2 = 09997

(b)






0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Actu

al re

cove

ry fa

ctor

(RF)

Data index


(a)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

Actu

al re

cove

ry fa

ctor

(RF)

Data index

(b)


0

2

4

6

8

10

0 10 20 30 40 50 60

Dev

iatio

n (

)



minus10

minus8

minus6

minus4

minus2




PSO-ANNRF NPV





012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)


012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)




02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2




119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160

Actu

al re

cove

ry fa

ctor

(RF)

Data index


(a)


0

10

20

30

40

50

60

0 10 20 30 40 50 60

Actu

al re

cove

ry fa

ctor

(RF)

Data index

(b)


0

2

4

6

8

10

0 10 20 30 40 50 60

Dev

iatio

n (

)



minus10

minus8

minus6

minus4

minus2




PSO-ANNRF NPV





012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)


012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)




02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2




119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










012345678

0 1 2 3 4 5 6 7 8

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(a)

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

PSO

-AN

N o

utpu

t

Net present value

DataFit R2 = 09996

(b)


012345678

0 20 40 60 80 100 120 140 160

Net

pre

sent

val

ue

Data index


(a)

01234567

0 10 20 30 40 50 60

Net

pre

sent

val

ue

Data index


(b)




02468

10

Dev

iatio

n (

)

minus10

minus8

minus6

minus4

minus2




119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877 =

sum119873

119894=1(119910119879

119894minus 119910119879) (119910

119875

119894minus 119910119875)

radicsum119873

119894=1(119910119879119894minus 119910119879)

2

sum119873

119894=1(119910119875119894minus 119910119875)

2

1198772= 1 minus

sum119873

119894=1(119910119879

119894minus 119910119875

119894)2

sum119873

119894=1(119910119879119894minus 119910119879)

2

MAE = 1

119873

119873

sum

119894=1

10038161003816100381610038161003816119910119879

119894minus 119910119875

119894

10038161003816100381610038161003816

MSE = 1

119873

119873

sum

119894=1

(119910119879

119894minus 119910119875

119894)2

(7)





5 Conclusions




Nomenclature

Abbreviations





Variables







References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Developing a Robust Surrogate Model of ...downloads.hindawi.com/journals/mpe/2015/706897.pdf · 3. Artificial Neural Network and Particle Swarm Optimization Arti