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Petroleum 1 (2015) 118e132

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Original article

Determination of oil well production performance usingartificial neural network (ANN) linked to the particle swarmoptimization (PSO) tool

Mohammad Ali Ahmadi a, Reza Soleimani b, Moonyong Lee c, Tomoaki Kashiwao d,Alireza Bahadori e, *

a Department of Petroleum Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iranb Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iranc School of Chemical Engineering, Yeungnam University, Gyeungsan, Republic of Koread Department of Electronics and Control Engineering, National Institute of Technology, Niihama College, 7-1 Yagumo-cho, Niihama, Ehime 792-8580,Japane School of Environment, Science & Engineering, Southern Cross University, Lismore, NSW, Australia

a r t i c l e i n f o

Article history:Received 15 May 2015Received in revised form6 June 2015Accepted 8 June 2015

Keywords:Well productivityDrainage areaSkin factorLeast square Support vector machineHybrid connectionist modelParticle swarm optimization

* Corresponding author.E-mail addresses: ahmadi6776@yahoo.com (M.A. A

scu.edu.au (A. Bahadori).Peer review under responsibility of Southwest Pe

Production and Hosting by Elsev

http://dx.doi.org/10.1016/j.petlm.2015.06.0042405-6561/Copyright © 2015, Southwest Petroleum Uaccess article under the CC BY-NC-ND license (http://

a b s t r a c t

Greater complexity is involved in the transient pressure analysis of horizontal oil wells in contrastto vertical wells, as the horizontal wells are considered entirely horizontal and parallel with the topand underneath boundaries of the oil reserve. Therefore, there is an essential need to estimateproductivity of horizontal wells accurately to examine the effectiveness of a horizontal well interms of technical and economic prospects.

In this work, novel and rigorous methods based on two different types of intelligent approachesincluding the artificial neural network (ANN) linked to the particle swarm optimization (PSO) toolare developed to precisely forecast the productivity of horizontal wells under pseudo-steady-stateconditions. It was found that there is very good match between the modeling output and the realdata taken from the literature, so that a very low average absolute error percentage is attained (e.g.,<0.82%). The developed techniques can be also incorporated in the numerical reservoir simulationpackages for the purpose of accuracy improvement as well as better parametric sensitivity analysis.

Copyright © 2015, Southwest Petroleum University. Production and hosting by Elsevier B.V. onbehalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND

license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Petroleum engineers are always interested in finding appro-priate and reliable tools to predict the productivity of horizontalwell as accurate predictions seem very important to conducttechnical and economical feasible studies before drilling

hmadi), alireza.bahadori@

troleum University.

ier on behalf of KeAi

niversity. Production and hostcreativecommons.org/licenses/b

the wells which is very costly. It is a very important factor todecide on the economical feasibility of drilling horizontal wells[1e7].

According to the experimental investigation conducted byYasar et al. [8]; applied load and torque for both specific energyand penetration rate in drilling operations have great impor-tance. Based on their results, penetration rate decreasesdramatically with raising Unconfined Compressive Strength(UCS) of the cement grout being drilled, proposing that rockcharacters will intensely effect advance in the drilling operation,and determined specific energy decreases dramatically withboth raising applied load and raising penetration rate in thedrilling operation [8]. Moreover, Wasantha et al. [9] studied theeffect of strain rate on the mechanical behavior of sandstoneswith various grain sizes. They concluded that size of constituentgrains is an essential factor which requires to be encompassed in

ing by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an openy-nc-nd/4.0/).

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132 119

thoughts of the mechanical behavior of sandstones undervarious strain rates [9].

The pseudo-steady state occurs when the fluid productionmanifests its impact in the boundaries far from the productionwell where the oil is approaching from the drainage boundary onthe way to the depletion area around the well [10,11].

It indicates that the reservoir has reached a condition wherethe pressure at all reservoir boundaries and also the averagereservoir pressure will reduce over time as more fluid isextracted from the reservoir [12,13].

A suitable technique to estimate the pressure transientresponse of producing horizontal wells was developed by Clontsand Ramey [14]; where the source functions are combinedwith the Newman product approach. Recently, an analyticalapproach was also introduced by Hagoort [15] to calculate theproductivity of a horizontal well with infinite conductivityplaced in a reservoir which is considered as a closed rectangularsystem.

The resultant of the source functions was utilized by Daviauet al. [16] for oil reservoirs with particular characteristics. Areliable solutionwas presented by Ozkan et al. [17] and Joshi [18]to estimate pressure response for the anisotropic and infinitereservoirs which experience either uniform influx or include ahorizontal well with infinite-conductivity.

Several researchers performed a variety of works to use thesource function scheme to obtain profile or/and magnitudes ofpressure in surrounded reservoirs [16,4,5,19,20,7,21].

The horizontal well-bore was considered as a strip source byGoode and Thambynayagam [22]. In their work a uniform dis-tribution of flow rate in the well length is maintained. Thepressure response was found for the horizontal producing wellthrough combination of the finite Fourier cosine and Laplacetransforms.

Depletionmode, especially in linear flow patterns depends onwell productivity in non-circular flow configurations. Helmy andWattenbarger [23] noticed that this observation matches wellwith the analytical long-time solutions for the analogous linearheat flow problems.

In general, the horizontal wells are not fully drilled in hor-izontal direction such that a significant change in verticalelevation happens during the drilling, leading to huge effects onoutlet pressure of the horizontal well. In addition, the compu-tation usually is difficult due to negative skin factor of hori-zontal wells. On the other hand, accurate calculation of thelength of horizontal production well is not simple in many cases[2].

Production at a constant well rate plays a major role in cur-rent analytical approaches to study the productivity of horizontalwells in closed reservoirs [24,25].

Helmy and Wattenbarger [26] presented simple productivitycorrelations using several number of reservoir simulations tocalculate the productivity of horizontal wells. The reservoirmodel contained a homogeneous, isotropic, closed reservoirwith the shape of a rectangular box and an open hole horizontalwell parallel to one of the sides of the box.

A semi-analytical model to determine of the productivity ofwells in Darcy flow systems was presented by Hagoort [27].

Finally, summary of the productivity models in steady stateand pseudo-steady state conditions is demonstrated in Table 1. Itshould be noted that, each of equations should only use in thespecific circumstances. In other words, various assumptionsshould be met to use each correlations (for instance, steady stateflow regime, homogeneous porous media and etc) and this is abig disadvantage for petroleum engineers to employ the afore-mentioned correlations. Moreover, most of the correlations

developed are depending on the size of the system because theyare not dimensionless and this is another drawback for usingthese formulas.

An attempt was made by Mutalik et al. [28] to compiledifferent data sets of shape factors and the corresponding skinfactors (SCAh), where different horizontal wells are considered atvarious locations of the drainage area in the reservoir [2]. Thepresence of an enclosed reservoir and an infinite-conductivitywell are the main assumptions in their methodology. The hori-zontal well productivity is determined by an equation as givenbelow [2]:

Jh ¼ qoPR � Pwf

¼ kh141:2bomo

0@ 1

ln�

r0erw

�� A0 þ Sf þ Sm þ SCAh � C0 þ Dqo

1A

(1)

where

r0e ¼ffiffiffiAp

r(2)

Sf ¼ �ln�

L4rw

�(3)

Bahadori et al. [29] introduced a predictive tool buy means ofVandermondematrix concepts, for estimating pseudo skin factorof horizontal wells through rectangular drainage area. Theynoticed that the calculation of productivity of a horizontal oilwell depends on the pseudo-skin factor for centrally locatedwells within different drainage areas.

Given the above matters, developing a proper and straight-forward correlation seems necessary. Compared to availableapproaches, the developed equation is expected to be lesscomplicated, and more accurate for predicting the pseudo-skinfactor as a function of dimensionless length (LD) and the ratioof horizontal well length over drainage area side (L/2Xe) forsquare and rectangular shapes with ratios of sides 1, 2, and 5.

LD ¼ L2h

ffiffiffiffiffiffiKv

Kh

s(4)

In this article, an intelligent method utilizing a new type ofnetwork modeling which called “Least Square Support VectorMachine (LSSVM)” is developed to serve as a rapid and inex-pensive predictive model for monitoring well productivity ofhorizontal wells in box shape drainage area. The proposedLSSVM model is developed implementing extensive actual wellproductivity data.

To depict the robustness, integrity and accuracy of the sug-gested LSSVM model, the obtained outcomes from the intro-duced approach are contrasted with the relevant actualproductivity data. Outputs from this research reveal that theevolved approach can monitor the horizontal well productivitywith high accuracy. The introduced predictive model can beutilized as a reliable way for quick and cheap but efficient pre-diction of well productivity of horizontal well parameter inabsence of appropriate experimental or/and real data, specif-ically through the initial stages of evolvement of horizontal welldrilling.

Table 1Summary of previous productivity equations.

Joshi-steady state [18] qo ¼ kHhðpe�pwf Þ

141:2moBo

8<:ln

24aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2�

�L2

�2s

L=2

35þIanih

L

�ln

�Ianih

rw ðIaniþ1Þ

�þs

�9=;

Butler-steady state [30] qo ¼ 7:08�10�3kHLðpe�pwf Þ

moBo

�Iani ln

�hIani

rw ðIaniþ1Þ

�þybp

h �1:14Ianiþs

�Furui et al.-steady state [10] qo ¼ 7:08�10�3kHLðpe�pwf Þ

moBo

�ln

�hIani

rw ðIaniþ1Þ

�þ ybp

hIani�1:224þs

�

Joshi-pseudo steady state [18]qo ¼

ffiffiffiffiffiffiffikxkz

pbðp�pwf Þ

141:2moBo

(ln

" ffiffiA

prw

#þln CH�0:75þsþsR

)

ln CH ¼ 6:28 aIanih

�13 � xo

a þ�xoa

�2�� ln

�sin pzo

h

�� 1

2 ln�

aIanih

�� 1:088

First radial [31]qo ¼ 2

ffiffiffiffiffiffiffiffikHkV

pL1=2ðpi�pwf Þ

162:6moBo

"log

" ffiffiffiffiffiffiffiffikHkV t

p∅rwCt rw2

#�3:2275þ0:8686s�2 log 1

2

ffiffiffiffikHkV

4q

þffiffiffiffikVkH

4q !#

t � 4moCt0:0002637pkV

minfz2w; ðh� zwÞ2g

Second radial [31]qo ¼

ffiffiffiffiffiffiffiffikHkV

pL1=2ðpi�pwf Þ

162:6moBo

"log

" ffiffiffiffiffiffiffiffikHkV t

p∅rwCt r

2w

#�3:2275þ0:4343s�log

" 1þ

ffiffiffiffikVkH

q !zwrw

##

4moCt0:0002637pkV

minfz2w; ðh� zwÞ2g � t � 4moCt0:0002637pkV

maxfz2w; ðh� zwÞ2g

Intermittent time-linear [31] qo ¼ ðpi�pwf Þ

Bo

"8:1282hLt

2

ffiffiffiffiffiffiffiffimo t

∅kHCt

qþ141:2mo

Khhszþ 70:6mo

Lt2

ffiffiffiffiffiffiffikHkV

p s

#

sz ¼ �1:1513ffiffiffiffikHkV

qhLt2

log

"prwh

1þ

ffiffiffiffikVkH

q !sin�pzwh

�#

4moCth2

0:0002637pkV� t � 4moCt L2t=2

0:0002637pkV

Late time-radial [31] qo ¼ ðpi�pwf Þ

moBo

"162:6khh

"log

" ffiffiffiffiffiffiffiffikHkV t

p∅rwCt r

2w

##þ141:2mo

Khhszþ 70:6mo

Lt2

ffiffiffiffiffiffiffikHkV

p s

#

t � 4moCtL2t=20:0002637pkV

sz ¼ �1:1513ffiffiffiffikHkV

qhLt2

log

"prwh

1þ

ffiffiffiffikVkH

q !sin�pzwh

�#� 0:5 kH

kVh2

L2t2

�13 � zw

h þ z2wh2

�

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132120

2. Description of physical model

Fig. 1 illustrates the schematic of the horizontal well config-uration which is characterized by the straightforward analyticalapproach: a cubic reservoir with a horizontal well parallel to thetop and bottom of the cube. The reservoir has a width w andthickness h and length 2Xe. An open hole horizontal well with alength of L is assumed drilled in the box shaped reservoir.

The location of the drilled well is determined by three pointswhich demonstrate the Cartesian coordinates such as xw, yw andzw. It should be noted that the Cartesian coordinates of the

Fig. 1. The schematic of the horizontal well in the box-shaped drainage area.

midpoint of the horizontal well throughout the Cartesian systemwhich is tier collateral to the edges of the cubic reservoir.Moreover, direction of the horizontal well is parallel to the x-axisof Cartesian coordinate system. The aforementioned reservoir isisotropic and homogeneous and consists a compressible fluidwith a constant compressibility and constant viscosity. No flowboundaries are assumed for the outer boundaries of the reser-voir. Production from well may be at constant pressure or con-stant rate. Well conductivity in aforementioned reservoir isconsumed infinite, which conveys pressure distributionthroughout the wellbore is uniform.

The 2D flow configuration of horizontal well in the above-mentioned system is depicted in Fig. 2. As clear be seen in thisfigure, two types of flow exist in horizontal wells includinghorizontal flow to a fully penetrating vertical fracture withlength Lw, and the flow to a fully penetrating horizontal well in arectangular reservoir of thickness h.

3. Methodology

3.1. Artificial neural network

The artificial neural networks (ANNs) have been built on thebasis of function and configuration of the brain of human, nervenetworks, and complicated procedure of learning and reacting. It

Fig. 2. Vertical and horizontal well configuration.

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132 121

aims to produce the magnitudes of target parameters from inputinformation through internal computations and analysis. Ingeneral, this connectionist technique involves well inter-connections model along with a simple processing constituent

Fig. 3. Structure of the proposed three laye

(or neuron) that is able to find out the right links between in-dependent and dependent parameters. The multilayer feed for-ward neural network is the most common and acceptable ANNsystem, where the neurons are coordinated into different layers;namely, input, output, and hidden.

Multilayer feed forward neural networks usually have one ormore hidden layers, which enable the networks to model non-linear and complex functions. The hidden layer is responsibleto convey meaningful information between the input andnetwork stages through an effective approach. ANNs interfacepredictors through the neurons of an input layer and transmitoutput of dependent variable(s) through the neurons of anoutput layer. Each neuron in hidden and output layerscarries a transfer function that illustrates internal activationphenomenon. A neuron output is normally achieved throughtransformation of its input employing an opposite transferfunction.

Transfer functions may be linear or non-linear. It should bementioned that, each interconnection in an ANN has a strengththat is expressed by a number referred to as weight. Further-more, bias is an extra input appended to neurons, and which allthe time holds a value of 1 and treated similar to other weights[32]. Fig. 3 illustrates the structure of a feed forward ANN withone hidden layer used in this study.

r feed forward neural network model.

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132122

A brief procedures to produce the output variable, using theinput data, are described below. Given the info for inputs neu-rons, the net inputs (S) for the hidden neurons are computed asfollows:

SHj ¼X2t¼1

wHj;t$at þ bHj (5)

where at is the vector of the input parameters, j is the index ofhidden neuron,wH

j;t denotes the interconnectionweight betweenthe input neurons with the hidden layer, and the term bHj standsfor the bias of the jth hidden neuron. Then, the outputs (Lj) of thehidden neurons are computed, utilizing a transfer function fHwhich is associated with the hidden neurons.

Lj ¼ fH�SHj�¼ fH

X2t¼1

wHj;t$at þ bHj

!(6)

The produced outputs from the hidden layer are presented tothe output layer as inputs. The output of the output layer (thefinal output, Y) can be also calculated in similar way. Indeed, it isresponsibility of all neurons in a network to relate the input in-formation to the desired output function through an adequateand suitable correlation.

3.2. Least squares support vector machines

Support vector machine (SVM) is a new kind of learningmachine which was developed within the context of statisticallearning theory and structural risk minimization by Vapnik andhis colleagues [33e36,37e41]. SVM is a very nice methodologywhich has recently attracted special attention from a great dealof researchers for solving problems in nonlinear classification,function estimation as well as for solving density estimationproblems [42,33e36,37e41], due to it enjoys several interestingproperties such as the use of the kernel-induced feature spaces,the sparseness of the solution, high generalization ability,together with excellent performance.

An evolved version of SVM, called LSSVM was recentlyintroduced by Suykens and co-workers in which the inequalityconstraints of an SVM were changed into a set of equality con-straints, in order to facilitate the original SVM method[33e36,37e41,53,44]. Therefore, the solution is obtained bysolving a system of linear equations, resulting in an easier-to-implement and faster alternative to the original SVM method[45e48,43,44]. It should be mentioned that, this section of thepaper serves as an overview to LSSVM. Details of this method canbe found in the literature [33e36,37e41,49,43,44].

Assume a set of N training data fðx1; y1Þ; ðx2; y2Þ;…; ðxN ; yNÞg,where xk2Rn is the kth input data and yk2R is correspondingoutput value. The aim is to estimate a model of the formula[33e36,37e41]:

yðxÞ ¼ W T4ðxÞ þ b (7)

where 4ð$Þ : Rn/Rnh is a nonlinear mapping function, whichmaps the input data to higher dimensional feature space; b is thebias term and W denotes the weight vector needed to bedetermined, which can be calculated by minimizing thefollowing function [33e36,37e41]:

J ðW ; eÞ ¼ 12W TW þ 1

2gXNk¼1

e2k (8)

Subject to the following equality constraint [33e36,37e41]:

yk ¼ W T4ðxkÞ þ bþ ek; k ¼ 1;2;…;N (9)

where g is the regularization constant used for preventing overfitting, and ek is the training data error. After then, the Lagrangianfunction is adopted as follow in order to solve the constrainedoptimization problem [33e36,37e41]:

L ðW ;b; e;aÞ ¼ J ðW ; eÞ �XNk¼1

ak

nW T4ðxkÞ þ bþ ek � yk

o(10)

where ak are Lagrange multipliers. By executing the first orderpartial derivatives of Lagrange function (8) with respect toW ;b; ek and ak the conditions for optimality can be written asfollow [33e36,37e41]:

vW L ¼ W �XNk¼1

ak4ðxkÞ ¼ 0 (11)

vbL ¼XNk¼1

ak ¼ 0 (12)

vekL ¼ ak � gek ¼ 0; k ¼ 1;…;N (13)

vakL ¼ 4ðxkÞ$W T þ bþ ek � yk ¼ 0; k ¼ 1;…;N (14)

When the variables W and e are removed, the optimizationproblems can be described as a linear system [33e36,37e41].

�0 1Ty1y Uþ g�1I

��ba

�¼�0y

�(15)

where y ¼ ½y1…yN �T, 1y ¼ ½1…1�T, a ¼ ½a1…aN �T, I is an identitymatrix and U is an N-dimensional symmetric matrix;Ukl ¼ 4ðxkÞT$4ðxlÞ ¼ Kðxk; xlÞ; ck; l ¼ 1;…;N. Kðxk; xlÞ is thekernel function and must meet Mercer's theorem. The resultingformulation of LSSVM model for function estimation is repre-sented as [33e36,37e41]:

yðxÞ ¼XNk¼1

akKðx; xkÞ þ b (16)

where (b, a) is the solution to the linear system (15).

4. Developed models

In the current research, we applied two types of artificial in-telligence based methodologies, viz. ANN and LSSVM to build thepredictive models to calculate the shape-related skin factor orpseudo-skin factor in horizontal wells as a function of dimen-sionless length and the ratio of horizontal well length overdrainage area side for various drainage areas based on previouslypublished database [2,28].

4.1. ANN model

4.1.1. Data distribution (training and testing subsets)When training multilayer feed-forward neural networks, the

general practice is to first separate the data into two subsets [45].The first subset is the training set, which is used for updating the

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132 123

network weights and biases. The accuracy of trained neuralnetwork can be independently measured by the second datasubset called testing set.

This will give us a sense of how well the network will dowhen applied to data from the real world. Herein, total of 96 datapoints (80% of whole data set) were used as training data andmain data set were considered as a test data.

4.1.2. Training method and transfer functionsA key consideration in ANN design is the type of transfer

functions. Different transfer functions can be used for neurons inthe hidden and output layers. Among different transfer functionsavailable, the log-sigmoid transfer function (logsig), which is anappropriate choice for many nonlinear functions, was employedas the transfer function for neurons in hidden layer which ismathematically expressed as:

f ðSÞ ¼ 11þ expð�SÞ (17)

The logsig function is bounded between 0 and 1. One of themost remarkable features of this transfer function lies in its Sshape which provides appropriate gain for a wide range of inputlevels. It should be mentioned that, ANNs be indebted their non-linear ability to exploit such non-linear transfer functions [50].

Also, for output layer, linear transfer function (purelin) isapplied as a transfer function which is represented as:

f ðSÞ ¼ S (18)

Outputs in the range of �∞ to þ∞, can be produced from thelinear transfer function [50].

ANNs' ability to deal with problems successfully greatly de-pends on applying an appropriate and effective training algo-rithm. Training process is basically an optimization process bywhich the weights of interconnections between neuronstogether with the biases are adjusted so that the network canpredict the correct outputs for a given set of inputs.

There are many different types of training algorithms whichhave been applied for various applications such as Back Propa-gation (BP) [45], Genetic Algorithm (GA) [51], Particle SwarmOptimization (PSO) [47,52,50,53], Hybrid Genetic Algorithm andParticle Swarm Optimization (HGAPSO) [54,55], Unified ParticleSwarm Optimization (UPSO) [56] and Imperialist CompetitiveAlgorithm (ICA) [57,58].

In this study, PSO algorithm was implemented as a trainingalgorithm for updating the weights and biases of multilayer feedforward network based on success stories of this new evolu-tionary computation algorithm for training ANNs in addition toits outstanding features including robustness, simplicity, flexi-bility, superior convergence characteristics and many others.

PSO is a population-based optimization algorithm based onthe simulation of the social behavior of bird flocks, fish schoolingand swarm of insects developed by [59], where the system israndomly initialized with a swarm of particles (potential solu-tions) and then searches the optima within the search spacethrough updating generations. Particles inside the swarm aredrawn progressively towards the global optimum as the systemiterates. In each iteration, each particle knows and stores the twobest values. The earliest value is the personal best position pi;pbest

that is the location of the particle i in the search space, where ithas attained the preeminent solution up to now, based on thefitness function. The global best solution pgbest is taken into ac-count as the second one that introduces a position, leading to thebest solution among all the particles on the same basis. Usingthis information, the velocity and position of each particle are

updated according to the following formulae and consequentlythe new swarm of particles is generated with improvedcharacteristics:

vnþ1i ¼ uvni þ c1r

n1

hpni;pbest

� pniiþ c2r

n2

hpngbest � pni

i(19)

pnþ1i ¼ pni þ vnþ1

i (20)

where vni and vnþ1i are velocities of particle i at iterations n and

n þ 1; pni and pnþ1i are positions of particle i at iterations n and

n þ 1; u is the inertia weight that plays an important role tohandle the exploration and exploitation of the search space, as itcontinuously corrects the value of velocity; c1 and c2 are termedas cognition and social components respectively are the accel-eration constants which changes the velocity of a particle to-wards pni;pbest

and pngbest , (generally somewhere between pni;pbestand

pngbest ) [60]; rn1 and rn2 are two random numbers uniformlydistributed in [0,1]; pngbest is the best position in the current swarmover generation n determined by the fitness function; and pni;pbest

is the best position of particle i over generation n determined bythe fitness function. A complete description and more details ofPSO can be found at a number of publications [60e65]. Herein,only the basic principles were briefly presented.

An opposite representation and fitness function (f) appear to benecessary before the PSO is utilized to train the connectionistnetwork model. Herein, neural network fitness was decided to bethemeansquareof errors (MSE)of thewhole trainingdata set [66]:

f ¼ 1N

XNi¼1

�yexpi � yprei

�2(21)

where N represents the total number of training data points(input and output pairs), yexpi is the actual (experimental) at thesampling point i, yprei is the ith output of the network. Everyparticle characterizes a possible solution to the optimizationcase. As biased of a trained neural network and the corre-sponding weights are counted a solution, one comprehensivenetwork is represented by a single particle.

Each component of a particle's position vector represents oneneural network weight or bias [54]. When the pre-specifiedcriterion is reached, the iterations cease and the set of theseoptimized weights and biases found is finalized and used in theproposed architecture (see Fig. 3). The detailed flowcharts of theimplementations of PSO algorithm for training the proposedANN architecture are shown in Fig. 4.

4.1.3. ANN structureNetwork structure is another important issue that needs

consideration due to its considerable impacts on the estimatedvalues. The number of input and output neurons corresponds tothe number of input and output data, respectively (8 and 1 in thisstudy). However, the most favorable numbers of hidden layersand neurons in each separate layer are entirely dependent on thecomplexity of the problem being solved by the neural networkand there is no a straightforward scheme to obtain theseparameters.

Hornik et al. showed that multilayer feed forward neuralnetworks consist of only one hidden layer with enough numberof hidden neurons can map any input to each output to anarbitrary degree of accuracy [67]. Based on the above, in thecurrent study only networks with one hidden layer was exam-ined. In order to find optimum number of neurons, different2 � x � 1 architectures (x varies from 1 to 10) were investigated.

Fig. 4. Flowchart of PSO-based optimization algorithm for evolving the weights and biases of the constructed ANN.

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132124

MSE and R2 values evaluated for different neural network con-figurations of varying number of neurons in the hidden layer areshown in Fig. 5.

The best configuration for the smart system is normallydetermined based on the minimum value of MSE and maximummagnitude of R2. As Fig. 5 depicts, the most appropriate ANNmodel consists of eight hidden neurons in one layer. Thus, it isfound that a three-layer network with a 2 � 8 � 1 differentprocessing stages is the best structure. The final trained ANNmodel with 8 hidden neurons is shown in Fig. 3.

Table 2 presents the adjustable parameters for the proposedhybrid PSOeANN approach.

Fig. 5. Effect of number of hidden neurons on performance of the PSOeANN modelin terms of R2 and MSE values.

4.2. LSSVM model

4.2.1. Data distribution (training and testing subsets)Like ANNs, for the LSSVM based-model analyses, the database

is broken down into training and testing subsets. The training

Table 2Details of trained ANNs with PSO for the prediction of pseudo-skin factor.

Type Value/comment

Input layer 2Hidden layer 8Output layer 1Hidden layer activation function LogsigOutput layer activation function PurelinNumber of data used for training 96Number of data used for testing 24Number of max iterations 500c1 and c2 in Eq. (15) 2Number of particles 20

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132 125

data are used for developing the model involves determining ofparameters embedded in the LSSVM model. The testing data areused tomeasure the performance of themodel and consequentlyto evaluate the precision and the effectiveness of the trainedLSSVM model on data that play no role in building it. For theanalysis, out of 120 data points, 96 data sets are taken for thetraining process and the rest of the data sets are used for testingthe predictive capability of the model.

4.2.2. Kernel functionIt is important to choose an appropriate kernel function.

There are many types of kernels; however, three typical choicesfor the kernel function are:

� Kðx; xkÞ ¼ xTkx ðLinear kernelÞ� Kðx; xkÞ ¼ ðtþ xTkxÞd ðPolynomial kernel of degree dÞ�Kðx; xkÞ ¼ expð�x� x2k=s

2Þ ðRadial basis function RBF kernelÞ

Entire magnitudes of s and positive extents of t meet theMercer condition [43]. There are many studies in the literaturewhich have made comparisons between the most commonkernels [45e47,43]. Provided nonlinear function prediction andnonlinear modeling cases, one can safely apply the RBF kernel as

Fig. 6. Flowchart of GA-based optimization algorithm to

it is an effective way that can produce accurate results[45e47,49,68,48]. Based on the above, this study employed theradial basis function (RBF) as a kernel function.

4.2.3. Optimization method to tune the embedded parameters (gand s2)

It is required to obtain the optimum values of kernel andregularization parameters through the training stage of theLSSVM technique. One effective way to conduct this task islinking the LSSVM model to genetic algorithm (GA) techniquewhich is employed in this study. Indeed, GA which has beeninitially developed by Holland [69] is recognized as a strongevolutionary algorithm to optimize Kernel RBF parameter (s2)and regularization parameter (g). This tool can be properlydesigned to handle a variety of optimization problems in thecontext of biological processes. GA operates with a collection ofcandidate solutions, called a population.

The system guides the candidate solutions towards an opti-mum using the principles of evolution and natural genetics. Asthe search process advances, the population incorporates betterand better individuals, and ultimately the convergence isattained, revealing that a single solution defeats it [69]. There arethree key operators including mutation, selection, and crossoverutilized at every generation of the GA model. It should bementioned that a chromosome corresponds to a solution vectorin GA. An extensive information on GA is found in the literature[65,61,69e72].

The flowchart of GA-based LSSVM incorporating to tune g ands2 values is shown in Fig. 6. It should be noted that the hybrid-ization of LSSVMmethodwith GA had already given encouragingresults for various applications [45e48]. The hybrid GAeLSSVMmodel searching mechanism for optimizing g and s2 parametersis briefly described as follows.

a. Encoding and generating Initial population: Defining a chro-mosome or an array of variable values to be optimized.Herein, the chromosome has 2 variables (a two-dimensionaloptimization problem) given by {g, s2}. Note that, a

adjust the embedded parameters of LSSVM model.

Fig. 7. Variation of skin factor versus dimensionless length (LD) at differentdrainage area (Xe/Ye) for L/2Xe ¼ 0.2.

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132126

chromosome is attributed to a unique solution in the solutionspace. Therefore, a mapping practice between the chromo-somes and the solution space is needed. Encoding is anothername for this mapping [69]. After representation of candidatesolutions, an initial population of chromosomes wasrandomly generated.

b. Fitness assignment: Evaluate the fitness of each chromosomein the population using a fitness function. Herein, the meansquared error of all training data set was established as fitnessfunction.

c. Selection: This stage involves repeating the most triumphantindividuals available in a population with a rate which isproportional to their relative quality. Indeed, chromosomeswith better fitness values have a greater chance of beingselected than those having worse fitness values.

d. Crossover: Two different solutions are putrefied in this stepand then the parts are randomly combined to create newsolutions.

e. Mutation: The process disturbs a possible solution through arandom way.

f. Replace: Use new generated population for the nextgeneration.

g. Stop criterion: The procedure is continued unless a number ofstopping conditions (e.g. when an acceptable solution hasbeen found or the number of generations that the GA isallowed to execute) are satisfied.

According to the hybrid GAeLSSVM developed in the currentstudy, the final magnitudes of g and s2 are determined to be2,158,321.5215 and 22.210787768, respectively; implying highpredictive potential of the model to determine the outputs withreliable and proper precision. Table 3 represents various settingsin adjustment of the LSSVM model and GA optimizationparameters.

5. Results and discussions

5.1. LSSVM output results

Before launching the outcomes of the developed vector ma-chine model for pseudo skin factor in rectangular drainage area,it is worth to mention that, distributions of the addressed target(pseudo skin factor) versus independent variables such asdimensionless length, L/2Xe and Xe/Ye are demonstrated in Fig. 7.Fig. 7 depicts the variation of pseudo skin factor versus corre-sponding dimensionless length at different drainage area for L/2Xe ¼ 0.2.

A regression plot between the estimated pseudo skin factorvalues and the real values are illustrated in Fig. 8. Fig. 8 dem-onstrates the scatter diagram that compares experimentalpseudo skin factor against LSSVM approach solutions. A tight

Table 3Basic parameter values of GAeLSSVM model for the prediction of pseudo-skinfactor.

Type Value/comment

Input layer 2Output layer 1Kernel function RBF kernel functionNumber of data used for training 96Number of data used for testing 24GA Population size 1000Max. number of generations 1000Crossover rate 0.8Mutation rate 0.02

cloud of points about diagonal line (Y ¼ X) for training andtesting data sets present the high performance of the developedLSSVM approach.

The addressed figure depicts that superior agreement existbetween the gained results of LSSVM model and the actualpseudo skin factor in rectangular drainage area. The statisticalcriteria of the evolved LSSVM approach which includes meansquared errors (MSE), average absolute relative deviations(AARD), and determination coefficients (R2) are summarized toTable 4.

Fig. 9 depicts the contrast of the measured values of pseudoskin factor and gained outputs of the suggested low parametermodel versus corresponding data index. As shown in Fig. 9, themodel outputs are reliable because follow exactly the measuredpseudo skin factor.

Fig. 10 illustrates the distribution for the deviation of gainedoutputs applying the LSSVM model versus experimental valuesof pseudo skin factor for all of the 120 data sets that imple-mented for developing the LSSVM approach. As demonstrated inFig. 10, maximum relative deviations of obtained results fromcorresponding actual skin factor is about ±5%.

Fig. 8. Regression plot of the proposed vector machine model versus actual pseudoskin factor.

Fig. 10. Relative error distribution of the obtained outputs from LSSVM modelversus corresponding pseudo skin factor data points.

Table 4Statistical parameters of the evolved LSSVM approach.

Training setR2 0.9988Average absolute relative deviation 0.77363Mean square error 0.14151N 96

Test setR2 0.9991Average absolute relative deviation 0.98169Mean square error 0.1445N 24

TotalR2 0.9983Average absolute relative deviation 0.81524Mean square error 0.14213N 120

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132 127

Moreover, Fig. 11 depicts the comparison between estimatedskin factor by utilizing our LSSVM model and actual skin factorversus relevant dimensionless length when L/2Xe ¼ 1. As illus-trated in Fig. 11, the obtained results from our LSSVM model arematched greatly with corresponding real skin factor for thisspecific system. In addition, Fig. 12 demonstrates the comparisonbetween estimated skin factor values by LSSVM model andactual skin factor in square drainage area (Xe/Ye ¼ 1) versusrelevant dimensionless length (LD). As shown in Fig. 12, LSSVMresult's followed real skin factor values with high level of pre-cision and accuracy.

To determine relative importance of the used input variableson the desired target of this study that is pseudo skin factor, arigorous statistical method has been implemented. Theaddressed technique named “analysis of variance (ANOVA)”. Theobtained materials from the mentioned statistical method areexhibited in Fig. 13. As exhibited in Fig. 13, dimensionless length(LD) has the highest positive effect on the pseudo skin factor ofhorizontal wells in the rectangular drainage area.

5.2. PSOeANN output results

A regression plot between the estimated pseudo skin factorvalues and the real values are demonstrated in Fig. 14. This figureillustrates the scatter diagram that contrasts real pseudo skinfactor against LSSVM approach solutions. A tight cloud of points

Fig. 9. Comparison between actual pseudo skin factor and predicted values byLSSVM model versus relevant data index.

about diagonal line (Y ¼ X) for training and testing data setspresent the high performance of the developed LSSVM approach.

The addressed figure depicts that superior agreement existbetween the gained results of LSSVM model and the actualpseudo skin factor in rectangular drainage area. Fig. 15 depictsthe contrast of the measured values of pseudo skin factor andgained outputs of the suggested hybridized model versus cor-responding data index. As shown in Fig. 15, the model outputsaren't reliable in all boundaries because correlation coefficientbetween measured pseudo skin factor and PSOeANN results ismuch lower than suggested LSSVM model.

Fig. 16 illustrates the distribution for the deviation of gainedoutputs applying the PSOeANN model versus experimentalvalues of pseudo skin factor for all of the 120 data sets thatimplemented for developing the PSOeANN approach. Asdemonstrated in Fig. 16, maximum relative deviations of ob-tained results from corresponding actual skin factor is about±30% that is not acceptable for engineering applications.

Moreover, Fig. 17 depicts the comparison between estimatedskin factor by utilizing our PSOeANN model and real skin factorversus relevant dimensionless length when L/2Xe ¼ 1. As illus-trated in Fig. 16, the obtained results from our PSOeANN modelaren't matched precisely with relevant real skin factor for this

Fig. 11. Comparison between predicted skin factor values by LSSVM model andactual skin factor when L/2Xe ¼ 1, versus relevant dimensionless length (LD).

Fig. 14. Regression plot of the proposed PSOeANN model versus actual pseudo skinfactor.

Fig. 12. Comparison between predicted skin factor values by LSSVM model andactual skin factor in square drainage area (Xe/Ye ¼ 1) versus relevant dimensionlesslength (LD).

Fig. 15. Comparison between actual pseudo skin factor and predicted values byLSSVM model versus relevant data index.

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132128

particular system. Furthermore, Fig. 18 depicts the comparisonbetween estimated skin factor values by PSOeANN model andreal skin factor in square drainage area (Xe/Ye ¼ 1) versus cor-responding dimensionless length (LD).

As shown in Fig. 18, results of PSOeANN model aren't fol-lowed real skin factor values with adequate level of precision andaccuracy. Finally, the comparison between statistical criteria ofthe proposed LSSVM model and PSOeANN model that containsmean squared errors (MSE), average absolute relative deviations(AARD), and determination coefficients (R2) are summarized toTable 5. As reported in Table 5, suggested LSSVM approach isrobust and effective in contrast with PSOeANN model in esti-mation pseudo skin factor of horizontal wells in rectangulardrainage area.

5.3. Leverage approach

Determining of the outlier of the mathematical approachesplays a vital role in applicability of the addressed approach in the

Fig. 13. Relative importance of each input variables on the pseudo skin factor in boxshape drainage area.

considered issue. Recognition of outlier means that the data,which are very different from the main points in the data bank,are identified. Hence, it seems vital to assess the real dataexisting for pseudo skin factor data. Since uncertainties influence

Fig. 16. Relative error distribution of the obtained outputs from PSOeANN modelversus corresponding pseudo skin factor data points.

Fig. 17. Comparison between predicted skin factor values by PSOeANN model andactual skin factor when L/2Xe ¼ 1, versus relevant dimensionless length (LD).

Table 5Calculated statistical indexes of the implemented intelligent based approachesfor pseudo skin factor determination.

Statistical parameter LSSVM PSOeANN

(MSE) 0.14213 1.5259R2 0.9991 0.9114AARD 0.81524 12.1011

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132 129

the estimation capability of the evolved approaches. To gain thisgoal, the approach of Leverage Value Statistics has been carriedout. The Graphical discovery of the doubtful data (or outliers) isconducted through theme of the Williams plot owing to thedetermined H values from the gained outcomes.

A detailed explanation of mathematical backgrounds andcomputational procedure of this approach can be found in pre-vious literature. The Williams plot is demonstrated in Fig. 19 forthe results implementing the vector machine and PSOeANNapproaches. It can be concluded that the developed techniquestatistically demonstrates high performance in terms of accuracyand applicability range as a large percentage of the data areplaced in the intervals H [0, 0.149] and R [�3, 3]. In addition, it

Fig. 18. Comparison between predicted skin factor values by PSOeANN model andactual skin factor in square drainage area (Xe/Ye ¼ 1) versus relevant dimensionlesslength (LD).

illustrates that the all data considered for the modeling approachare present within the acceptable ranges.

6. Conclusions

In this paper, an LSSVM model for predicting pseudo skinfactor of horizontal wells in the rectangular drainage area hasbeen developed. Through this work massive horizontal wellproductivities data bank's [1, 21] gathered from literature sur-veys are faced to the LSSVM model to evolve and examine itsrobustness. Following deductions can be drawn based on solu-tions obtained from the LSSVM and PSOeANN models:

1. Without any doubts at all which having adequate qualitativeand quantitative expertise about pseudo skin factor as

Fig. 19. Detection of the probable doubtful measured skin factor data [1, 21] and theapplicability domain of the suggested approaches for the pseudo skin factor. The H*

value is 0.125.

M.A. Ahmadi et al. / Petroleum 1 (2015) 118e132130

fundamental factor which plays the leading role in produc-tivity of the horizontal wells. In order to gain economic andtechnical optimizing goals, a numerous number of conven-tional studies have been done to propose some models forprecise calculation of this parameter, but some intrinsic lim-itations and constraints have triggered attentions to be drawntowards numerical intelligent models.

2. Based on the employed statistical indexes the monitoring ofthe pseudo skin factor of horizontal wells made by developedLSSVM approach result the closest arrangement with the realdata among the artificial neural network approaches.

3. The summary is that performing the evolved approach causesreaching a high degree of performance and precision in termsof calculating the pseudo skin factor of horizontal wells thathas not already been conceivable via using conventionalschemes.

4. The bottom line of this paper is that the LSSVM model fordetermining the pseudo skin factor of horizontal wells in box-shaped drainage area is easy-to-use model for petroleumengineers. Moreover, the aforementioned model can couplewith commercial reservoir simulators to improve accuracyand decrease run time. Finally, the LSSVM model proposed inthis paper can aim the petroleum engineers to determineoptimum well location with the concept of reverseengineering.

Nomenclature

AbbreviationsAARD average absolute relative deviationsANN artificial neural networkANOVA analysis of varianceBP back propagationEOS equation of stateGA genetic algorithmHGAPSO hybrid genetic algorithm and particle swarm

optimizationICA imperialist competitive algorithmLSSVM least square support vector machineMAE mean absolute error (MAE)MSE mean squared error (MSE)PSO particle swarm optimizationR2 coefficient of determinationRBF radial basis functionUPSO unified particle swarm optimization

Variables

yP the average of the predicted data

yT the average of the actual dataA drainage area of horizontal well, ft2

A extension of drainage volume of horizontal well in x-direction, ft

Af area open to flow, ft2

b extension of drainage volume of horizontal well in y-direction, ft

bj biasBo oil formation volume factor, rb/STBC unit conversion factorc1 cognition componentc2 social componentsCH geometric factor in Babu and Odeh's modelct total compressibility, psi�1

D non-Darcy flow coefficient, day/Mscf

e error ¼ Actual � Model outputh reservoir thickness, ftIani permeability anisotropy, dimensionlessJ productivity index, STB/day/psiK effective permeability, mdkd effective permeability in damaged zone, mdkH horizontal permeability, mdkV vertical permeability, mdkx permeability in x-direction, mdky permeability in y-direction, mdkz permeability in z-direction, mdL horizontal wellbore length, ftL1/2 half-length of horizontal wellbore, ftLD dimensionless lengthN the total number of data pointsoj outputp reservoir pressure, psip0 reference pressure, psipavg average reservoir pressure, psipe reservoir pressure at boundary, psipi initial pressure, psipwf bottom-hole flowing pressure, psiqo oil production rate, STB/dayrn1 and rn2 two random numbersrdH radius of damaged zone in horizontal direction, ftre radius of outer boundary of the reservoir, ftrw wellbore radius, fts skin factor, dimensionlessSj sum of interconnection weightssR partial penetration skin factor, dimensionlesst time, hvi velocity of particle iWji interconnection weights in network modelx0 x coordinate of center of well, ftxi position of particle ixe half of drainage area side, ftyPi the ith output of the modelyTi the actual at the sampling point iz0 z coordinate of center of well, ftzw distance of wellbore from the lower boundary, ft

Greek lettersm viscosity, cpmo oil viscosity, cpØ porosity, dimensionlessg regularization parameterd absolute relative errorr density, lbm/ft3s2 RBF parameter4 activation functionu the inertia weight

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