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Petroleum 3 (2017) 242e248

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Petroleum

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Decline curve based models for predicting natural gas wellperformance

Arash Kamari a, Amir H. Mohammadi a, b, c, *, Moonyong Lee d, Alireza Bahadori e, f, **

a Disciples of Chemical Engineering, School of Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban4041, South Africab Institut de Recherche en G�enie Chimique et P�etrolier (IRGCP), Paris Cedex, Francec D�epartement de G�enie des Mines, de la M�etallurgie et des Mat�eriaux, Facult�e des Sciences et de G�enie, Universit�e Laval, Qu�ebec, QC G1V 0A6,Canadad School of Chemical Engineering, Yeungnam University, Gyeungsan, Republic of Koreae School of Environment, Science & Engineering, Southern Cross University, Lismore, NSW 2480, Australiaf Australian Oil and Gas Services Pty Ltd, Lismore, NSW 2480, Australia

a r t i c l e i n f o

Article history:Received 31 December 2015Received in revised form22 June 2016Accepted 27 June 2016

Keywords:Decline curveProductionRateModelGas well

* Corresponding author. Thermodynamics Researneering, University of KwaZulu-Natal, Howard CollegAvenue, Durban 4041, South Africa.** Corresponding author. School of Environment,Southern Cross University, Lismore, NSW 2480, Austr

E-mail addresses: a.h.m@irgcp.fr, amir_h(A.H. Mohammadi), alireza.bahadori@scu.edu.au (A. B

Peer review under responsibility of Southwest Pe

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http://dx.doi.org/10.1016/j.petlm.2016.06.0062405-6561/© 2016 Southwest Petroleum University. Punder the CC BY-NC-ND license (http://creativecomm

a b s t r a c t

The productivity of a gas well declines over its production life as cannot cover economic policies. Toovercome such problems, the production performance of gas wells should be predicted by applyingreliable methods to analyse the decline trend. Therefore, reliable models are developed in thisstudy on the basis of powerful artificial intelligence techniques viz. the artificial neural network(ANN) modelling strategy, least square support vector machine (LSSVM) approach, adaptive neuro-fuzzy inference system (ANFIS), and decision tree (DT) method for the prediction of cumulative gasproduction as well as initial decline rate multiplied by time as a function of the Arps' decline curveexponent and ratio of initial gas flow rate over total gas flow rate. It was concluded that the resultsobtained based on the models developed in current study are in satisfactory agreement with theactual gas well production data. Furthermore, the results of comparative study performed dem-onstrates that the LSSVM strategy is superior to the other models investigated for the prediction ofboth cumulative gas production, and initial decline rate multiplied by time.© 2016 Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf 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

Hydrocarbon reserves estimation and/or production perfor-mance forecasting is regarded as one of the most challengingtasks in petroleum engineering disciplines. To this end, a reliable

ch Unit, School of Engi-e Campus, King George V

Science & Engineering,alia._mohammadi@yahoo.comahadori).troleum University.

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roduction and hosting by Elsevons.org/licenses/by-nc-nd/4.0/)

method called decline curve is used to fit recorded productionrate related to individual and group of wells or reservoirs by amathematical/computational function for forecasting the pro-duction performance by conducting the extrapolation of thedecline function fitted [1]. To forecast the well or reservoir pro-duction performance, manymodels have been proposed by usingdifferent approaches. Xu et al. [2] proposed an analytical modelto analyse the production data of shale gas and tight gas reser-voirs. The solutions made by them are applied for analysing thetype curve of production data for both homogeneous and natu-rally fractured reservoirs. Furthermore, they reported a new typecurve to evaluate the production performance of shale and tightgas reservoirs. They reported that there is satisfactory agreementbetween the results obtained based on the developed analyticalmodel and the numerical methods to which it was compared.

Zhang et al. [3] developed a composite method for analysingthe decline performance of a multiple fractured horizontal wellin shale gas reservoirs. They concluded that their developed

ier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article.

A. Kamari et al. / Petroleum 3 (2017) 242e248 243

composite model can provide some visions into the mechanismof fluid flow in shale gas reservoirs, and help to analyse thedecline production performance of such wells and reservoirs.Fanchi et al. [4] analysed the probabilistic decline curve work-flow for modelling the production data of the some shale gasreservoirs. Khanamiri [1] presented two semi theoreticalmethods for both oil and gas reservoirs on the basis of time andrate variables. The results indicated that the oil reservoir modeloutputs are same with those obtained by hyperbolic equation,and moreover the gas reservoirs model outputs are reliable incomparison with the Arp's method [5].

Regarding the importance of predicting the production per-formance of hydrocarbon reservoirs, development of easy-to-use, reliable, and accurate models is of much important. To thisend, intelligent based techniques show better capability formodelling nonlinear problems than the traditional methods suchas empirically derived correlations as well as production chartsso that this makes it more attractive for prediction of productionwell performance. In comparison with the intelligent basedtechniques, the empirically derived correlations as well as pro-duction charts have a poor performance for prediction targets inthe presence of a small size of dataset because of their highnumber of adjustable parameters. Therefore, the empiricallyderived correlations as well as production charts may have theover-fitting problem so that they may lead to error in the outsideranges of data which have been developed based on [6]. In otherwords, the empirically derived correlations and productioncharts may be applicable for analysing the performance of justsome production wells with specific data while intelligent basedmodels are powerful tools which can predict production wellperformance in any range of data associated with variousgeographical locations.

Therefore, application of artificial intelligence approaches viz.artificial neural networks, fuzzy logic, decision tree (DT), andleast square support vector machine, etc., is of helpful to proposesuch models. Artificial intelligence methods have fruitfully beenimplemented in predicting the thermodynamics of petroleumreservoir fluids, oil production and optimization, enhanced oilrecovery, petroleum flow assurance, and reservoir engineering[7e9]. As a result, the artificial neural network (ANN) modellingstrategy optimized with particle swarm optimization (PSO), leastsquare support vector machine (LSSVM) approach coupled withthe coupled simulated annealing (CSA) optimization method,adaptive neuro-fuzzy inference system (ANFIS), and finally DTmethodwere implemented in current study to accurately predictthe cumulative gas production and initial decline rate multipliedby time as a function of the Arps' decline curve exponent andratio of initial gas flow rate over total gas flow rate.

2. Database

As a result, the decline curve model proposed by Arps [5] isstill considered as one of most reliable methods for predictinghydrocarbon reserves estimation and production performancerelated to oil and gas wells as well as reservoirs so that almost allconventional decline curve methods/models are developed onthe basis of Arps' model as follows [10]:

qt ¼ qið1þ bDitÞ1=b

(1)

where qt (MMscf/day) stands for the total gas flow rate at time t,qi (MMscf/day) denotes the initial gas flow rate, Di (1/day) ex-presses the initial decline rate, and b indicates the Arps' declinecurve exponent.

Regarding the matters discussed above, the cumulative gasproduction ((GP2-GP1)/(tqi)) and initial decline rate multipliedby time (Dit) as a function of the Arps' decline curve exponentand ratio of initial gas flow rate over total gas flow rate have beenmodels based on the data taken from the literature [11,12]. Thedata used in this study are taken from decline curve analysischarts relating production rate to time as well as cumulativeproduction reported by Gentry [11]. Gentry [11] tested the chartsby the results of production history of a Mississippi limestonewell in Northwestern Oklahoma with initial production rate (qi)6292 bbl/month; production rate at time t (qt) 730 bbl/month;cumulative production at time t (Qt) 55,960 bbl; ultimate pro-duction (Qultimate) 104,000 bbl; and a production time 27months(time between qi and qt). Ranges (i.e. maximum, minimum andaverage) of the cumulative gas production and initial decline ratemultiplied by time as outputs, as well as the Arps' decline curveexponent and ratio of initial gas flow rate over total gas flow rateas input variables are listed in Table 1.

3. Methodologies for model development

3.1. Artificial neural network e multilayer perceptron (MLP)model

Artificial neural network is an advanced model to process andclassify information which simulates the biological neuralnetwork in human brain and is based on themathematical systemof processes which happens in brain [13e15]. Multilayer percep-tron (MLP) denotes the advanced type andwell researched type ofANNs in regression and classification, applying a feed forward,supervised and hetero-associative model [16]. The MLP andmanyother neural networks use an algorithmnamed back-propagation.By using the back-propagation, the input data is repetitively im-ported to the neural network. With each steps that outputs aregenerated, the values will be compared to measured outputs andconsequently error is calculated. This error is then provided back(back-propagated) to the ANN and used to regulate the weightssuch that the error declines with each iteration and the neuralmodel produces accurate outputs which are close to desired tar-gets. This procedure is recognized as “training” [17].

In artificial neural networks, information handling isexecuted in numerous simple separate processors which areentitled neurons. When inputs are imported to the neuron, theyare multiplied by related weight of each link. Then bias(threshold) of the neuron is supplied to summation of weightedinputs. In artificial neural network approach, the data from theinput neurons are propagated through the network via weightedinterconnections [18]. Every i neuron in a k layer is connected toevery neuron in adjacent layers. The activation function ofexponential sigmoid function which has generally and tradi-tionally been utilized to develop ANNs [19] as following:

f ðxÞ ¼ 11þ e�x (2)

where x stands for parameter of activation function. A bias term,b0, is associated with every interconnection in order to introducea supplementary degree of freedom. The expression of theweighted sum, S, to the ith neuron in the kth layer (k � 2) is [18].

Sk;i ¼XNk�1

j¼1

h�wk�1;j;iIk�1;j

�þ b0k;i

i(3)

where w is the weight parameter between each neuroneneuroninterconnection. Using this feed-forward network with

Table 1Ranges of data employed for the prediction of gas well performance using decline curve models developed in this study.

Variable Min. Avg. Max. Type

Arps' decline curve exponent 0 0.5 1 InputRatio of initial gas flow rate over total gas flow rate 0.2 3.0 10 InputCumulative gas production 0.7 7.8 40 OutputInitial decline rate multiplied by time 0.011 0.3 0.726 Output

A. Kamari et al. / Petroleum 3 (2017) 242e248244

activation function, the output, O, of the i neuron within thehidden k layer is [18].

Ok;i ¼1

1þ e��PNk�1

j¼1 ½ðwk�1;j;i Ik�1;jÞþb0k;i�¼ 1

1þ e�Sk;i(4)

3.2. Least squares support vector machine model

Least squares support vector machines (LSSVM) are leastsquares forms of support vector machines (SVM), which are a setof associated supervised learning methods that investigate dataand identify patterns, and which are used for sorting andregression analysis offered by Suykens et al. [20e22]. In LSSVM alinear approximation is prepared in kernel induced featurespace. By considering a data set (xi, yi), i ¼ 1, 2, … N, with inputdata xi ε R and output data yi ε R. The regression model can beestablished as follows [22e24]:

yk ¼ wT∅ðxkÞ þ bþ ek; K ¼ 1;2;…;N (5)

In these equations, w characterizes the linear regression(regression weight), T is symbolic of the transpose matrix, e istraining items regression error, b represents the model linearregression intercept, and ∅ shows the feature map. The costfunction of LSSVM algorithm, QLSSVM is calculated as below[21e23].

QLSSVM ¼ 12wTwþ g

Xnk¼1

e2k (6)

g is the relative weight of the regression errors summationcompared to the regression weight. By assistance of Lagrangefunction, the regression weight is normally shown as follows[25,26].

w ¼Xnk¼1

akxk (7)

In which ak is defined as.

ak ¼ 2gek (8)

With the assumption of linear regression between indepen-dent and dependent LSSVM variables, Eq. (5) can be re-written as[23,25].

y ¼Xnk¼1

akxTkxþ b (9)

With the subsequent equation the Lagrange multipliers, akcan be considered as [21,22,26].

ak ¼ðyk � bÞ

xTkxþ ð2gÞ�1 (10)

By means of Kernel function the first linear regression equa-tion will changed into a nonlinear form [24e26].

f ðxÞ ¼Xnk¼1

akKðx; xkÞ þ b (11)

In which K(x,xk) represent the Kernel function, made by innerproduct of the vectors ∅ðxÞ and ∅ðxkÞ [22,25].

Kðx; xkÞ ¼ ∅ðxÞT$∅ðxkÞ (12)

Radial basis function (RBF) is the utmost used relation forcalculating the Kernel function [23,26].

kðx; xkÞ ¼ exp�� k xk � xk2

s2

�(13)

Here s2 is a decision variable. Its optimization controlled byan external procedure during model's internal computations.The mean square error (MSE) definition for the LSSVM can bedefined as follows [25,26].

MSE ¼Pn

i¼1

�T rep

pred� Texp

�ns

(14)

Which T is the wax disappearance temperature, rep/pred andexp specify the represented/predicted and experimental datarespectively and ns is the initial population number [25,26].

3.3. Regression decision tree (DT) model

Decision tree constructs models in purpose of regression orclassificationwith the shape of a tree structure. It divides a datasetinto smaller subsections while a related decision tree is incre-mentally developed simultaneously. The outcome is a tree withdecision nodes and leaf nodes. A decision node has two or moredivisions, each signifying values for the feature tested. Leaf noderepresents a decision on the target. The top decision node in a treewhich relates to the best predictor is called root node. Decisiontrees can process both categorical and numerical data [27e29].There are three kinds of decision tree: CRT, CHAID and ExhaustiveCHAID, and Quest. The procedures of the three sorts follow thefollowing phases. Start tree building by allocating the node toclasses, stopping tree building. Reach the optimum tree selectionand perform cross-validation. CRT makes tree “Pruning” beforecreating the optimum tree selection, while CHAID method im-plements statistical tests at each step of splitting [30e32].

The CRT (Classification and Regression Tree) is a recursivesubdividing technique to be used both for regression and clas-sification. The CRT is built by division of data set into the sub-groups using all predictor variables to create two child nodesrepeatedly, beginningwith thewhole data set. The best predictoris chosen using a diversity measures. The objective is to yieldsubdivisions of the data which are as similar as possibleregarding to the target variable [33e35]. The CHAID (Chi-SquareAutomatic Interaction Detector) technique is established basedon the X2-test of association. A CHAID algorithm is a decision tree

A. Kamari et al. / Petroleum 3 (2017) 242e248 245

that is created by repetitively division subgroups of the spaceinto two or more child nodes. To control the best splitting at anynode, any acceptable couple of groups of the predictor variablesis combined till there is no statistically important alteration inthe couple with respect to the target variable. This CHAIDmethod logically works with communications between the in-dependent variables that are directly accessible from an analysisof the tree. The final nodes recognize subsections defined bydiverse groups of independent variables [35,36].

The original CHAID procedure is not assured to discover thebest split of all of those inspected because it uses the last splittested. The Exhaustive CHAID procedure tries to solve this issueby continuing to combine groups, regardless of significance level,until only two groups stay for each predictor. It then uses thesplit with the major importance value rather than the last onetried. The Exhaustive CHAID needs additional computer time[30,35]. The QUEST (Quick-Unbiased-Efficient Statistical Tree) isa binary split decision tree process for classification and dataanalysis. The QUEST can be used with univariant or lineargrouping splits. An exceptional characteristic is that its attributeselection technique has insignificant bias. If all the attributes areuninformative with respect to the class attribute, then each hasapproximately the identical variation of being selected to split anode time [30,35]. Finally, here it is worthwhile to note that aregression tree is similar to a classification tree, except that the Yvariable takes ordered values and a regression model is fitted toeach node to give the predicted values of Y [37].

3.4. Adaptive neuro-fuzzy inference system model

Theadaptiveneuro-fuzzy inference systemproposedby Jang in1993 [38] is viewed as a smart hybrid methodology composed of,and/or combining both fuzzy logic and artificial neural networks.The production of reasonable results by means of the imple-mentation of the simple fuzzy inference system (FIS) is directlyassociatedwith expert-knowledge rules. The absenceof such rulesis recognized as a major disadvantage of this intelligent system.

The definition of intelligent rules and appropriatemembershipfunctions can help to avoid imperfect results. The ANFIS combines

Fig. 1. Crossplots of the initial decline rate multiplied by time data versus the predictapproaches. Y axis represents the model results and X axis stands for actual data.

the strengths of fuzzy logic and artificial neural networks in orderto develop models on the basis of an FIS with optimized rules andmembership functions. The ANFIS methodology relies on theassumption that ANN trains the whole network and FIS structureto pursue the fuzzy logic representation. It is worth mentioningthat the structures of rules and functions in models developedbased on the ANFIS method are similar to those which have beenproposed by the concept of fuzzy logic [39].

4. Results and discussion

To forecast the cumulative gas production and initial declinerate multiplied by time, eight reliable models have been devel-oped on the basis of the ANN, LSSVM, ANFIS, and DT strategies. Tothis end, the same variables viz. the Arps' decline curve exponentand ratio of initial gas flow rate over total gas flow rate have beenconsidered as inputs to predict the gas well performance. Forvisualizing the precision performance and capability of themodels developed based on the methods mentioned above, areliable error parameter called average absolute relative devia-tion (AARD) is applied as follows:

AARD% ¼ 1n

Xni¼1

����Xexp � Xrep:=pred

Xexp

����� 100 (15)

As e result, the particle swarm optimization [40] strategy wasutilized in this study to tune the adjustable parameters (bias andweight) related to the feed-forwardANNalgorithm.Afterward, thenumber of hidden neurons in hidden layer has been optimized byconducting the try and error method. The most precise modelsdeveloped on thebasis of ANNstrategywere selectedwith respectto the AARD percent. Furthermore, the coupled simulatedannealing (CSA) [41] optimization strategy was used to tune theadjustable parameters of the LSSVM viz. g and s2. As a conse-quence, the values optimized by theCSAmethod ares2¼ 0.41 andg ¼ 634,776 for the model developed for prediction of the cu-mulative gasproduction, ands2¼ 5.4 andg¼1.8E11 for themodelproposed for estimating the initial decline ratemultiplied by time.Additionally, the regression DTalgorithm available in theMATLAB

ed values by the developed model based on the LSSVM, DT, ANFIS, and ANN-PSO

Fig. 2. Crossplots of the actual cumulative gas production data versus the predicted values by the developed model based on the LSSVM, DT, ANFIS, and ANN-PSO approaches.Y axis represents the model results and X axis stands for actual data.

A. Kamari et al. / Petroleum 3 (2017) 242e248246

was used to develop a predictive model for comparing the esti-mated data values with the other methods investigated in thisstudy. Finally, having tried a number of membership functions todevelop a reliable method on the basis of ANFIS method, theGaussian membership provided reasonable results.

To evaluate the accuracy performance of the models devel-oped in this study, the AARD % was calculated for each model.The cumulative gas production datawas predicted with an AARD% of 6.95, 8.95, 14.66, and 30.5% for the LSSVM, DT, ANFIS, andANN-PSO models, respectively. Moreover, the models proposed

Fig. 3. Absolute relative deviation contour of the predicted initial decline rate multipliedgas flow rate over total gas flow rate (qi/qt).

on the basis of the LSSVM, DT, ANFIS, and PSO-ANN could esti-mate the initial decline rate multiplied by time with an AARD %of 2.26, 10.5, 23.62, and 47.43%, respectively. As clear from theresults obtained, the LSSVM models proposed in current studyhave highest accuracy in predicting both cumulative gas pro-duction and initial decline rate multiplied by time. Moreover, thepredicted values by the regression DT model developed in thisstudy are in more agreement with the actual production data incomparison with the ANFIS and ANN-PSO models. Figs. 1 and 2illustrate the parity diagrams of the predicted values by all

by time in the different ranges of Arps' decline curve exponent (b) and ratio of initial

Fig. 4. Absolute relative deviation contour of the predicted cumulative gas production in the different ranges of Arps' decline curve exponent (b) and ratio of initial gas flowrate over total gas flow rate (qi/qt).

A. Kamari et al. / Petroleum 3 (2017) 242e248 247

methods investigated in this study for the initial decline ratemultiplied by time and cumulative gas production data,respectively. As clear from the figures, distribution of the pre-dicted values by the developed LSSVM models around the unitslope line is lowest compared to the other models studied.Additionally, Figs. 3 and 4 displays the absolute relative deviation(ARD) contour of the predicted gas well production data in thedifferent ranges of Arps' decline curve exponent and ratio ofinitial gas flow rate over total gas flow rate for the initial declinerate multiplied by time and cumulative gas production data,respectively. Fig. 3 demonstrates that all methods investigated inthis study have better performance in the higher range of b andqi/qt for estimating the initial decline rate multiplied by time.Additionally, the models developed in current study (LSSVM, DT,ANFIS, and ANN-PSO) are not recommended for the prediction ofcumulative gas production in the ranges of 0.7e0.9 for b and 3e5for qi/qt, as shown in Fig. 4.

5. Conclusions

The challenges faced with the petroleum engineers to esti-mate the hydrocarbon reserves, and predict the productionperformance of oil and gas wells were considered in this studywith development of reliable models for prediction of the cu-mulative gas production as well as the initial decline ratemultiplied by time. To this end, the Arps' decline curve exponentand ratio of initial gas flow rate over total gas flow rate have beenconsidered as inputs of the models. Reliable artificial intelligencetechniques viz. the regression decision tree, least squares sup-port vectormachine, adaptive neuro-fuzzy inference system, andneural network were employed to analyse the gas wells perfor-mance. It was concluded that the results obtained based on themodels developed in current study are in satisfactory agreementwith the actual gas well production data. Finally, a comparativestudy was performed between the all models developed in

current study and the actual gas well production data. It wasfound that the LSSVM strategy is superior to the other modelsinvestigated for the prediction of both cumulative gas produc-tion, and initial decline rate multiplied by time.

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