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Genetically-optimised neural networks

Genetic-adaptive neuro-fuzzy inference

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network and a genetically enhanced subtractive clustering for the parameter identication of an

adaptive neuro-fuzzy inference system. Simulation experiments are provided, showing the perfor-

mance of the proposed techniques as compared with commonly used regression correlations, including

sedimnt ofality rovoir-enof resessainynsist

other, more relatively abundant, measures. One example of such a

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performance compared to the use of each technique alone. A case

Contents lists available at SciVerse ScienceDirect

.el

Computers &

Computers & Geosciences 44 (2012) 109119function of temperature, oil gravity, gas gravity and gas/oil ratio,E-mail address: amar@kfupm.edu.sakind of measures is well logs, which are a series of multi-type study of two important PVT properties prediction problem isconsidered. These two properties are bubble point pressure (Pb),and oil formation volume factor (Bob). The former denes thepressure at which the rst gas comes out of solution in oil. It is a

0098-3004/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.cageo.2012.03.016

n Tel.: 966 3 860 7614; fax: 966 3 860 2965.well locations and/or intervals. There is a great demand for thedevelopment of correlation models to relate these properties to

explored and investigated by gathering their respective power-fulness in hybrid optimised structures, so as to improve theirThe determination of reservoir properties is however a com-plex problem, because laboratory-measured properties of rocksamples (cores) are only available from limited and isolated

neural networks, have been developed based on available dadifferent regions of the world.

In this paper, the capabilities of soft computing techniquewhich are used to characterise the spatially varied geologicalinformation. Fluid characterisation quanties the reservoir phasebehaviour, uid compositional changes throughout the reservoir,and changes in uid properties as a result of production andinjection processes.

To overcome the abovementioned problems of expensivelaboratory-based tests and the possible non-availability of theright samples from the well-bore or well surface, equations ofstate models and several empirically derived correlationsbetween those properties and well data, and eventually articial1. Introduction

1.1. Motivation and background

A reservoir is a volume of porousbeen lled with a substantial amoucrude oil and natural gas. High-qucritical for reliable modelling, reserand performance predictions by usesubsequent economic analysis (Al-Hu

Typically, reservoir properties co& 2012 Elsevier Ltd. All rights reserved.

entary rock, which hashydrocarbons, such asck and uid data aregineering calculations,rvoir simulators and forand Humphreys, 1996).of a set of parameters

digital measurements along the vertical depth of drilled wells.These models are used to transform the well log data intoreservoir properties at locations where no cores are available.

PressureVolumeTemperature (PVT) properties are key para-meters associated with the characterisation of any hydrocarbonreservoir. In fact, it is not possible to have accurate solutions tomany petroleum engineering problems without having accurateestimates of these properties. Furthermore, the control of therelationship between the surface and reservoir hydrocarbonvolumes and the underground withdrawal is made possible byknowing the oil PVT parameters.standard articial neural networks.Hybrid soft computing systems for rese

Amar Khoukhi n

Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dh

a r t i c l e i n f o

Article history:

Received 25 April 2011

Received in revised form

15 March 2012

Accepted 18 March 2012Available online 1 April 2012

Keywords:

PressureVolumeTemperature

Oil formation volume factor

Bubble point pressure

Correlation

a b s t r a c t

In reservoir engineering, t

importance for many uses

inow performance calcul

the potential developme

properties is a complex p

are only available from lim

have been developed to re

models include empirical

this paper, a comprehensi

formation volume factor u

journal homepage: wwwoir PVT properties prediction

n, 31261, Saudi Arabia

nowledge of PressureVolumeTemperature (PVT) properties is of great

h as well test analyses, reserve estimation, material balance calculations,

ns, uid ow in porous media and the evaluation of new formations for

and enhancement oil recovery projects. The determination of these

lem because laboratory-measured properties of rock samples (cores)

d and isolated well locations and/or intervals. Several correlation models

these properties to other measures which are relatively abundant. These

relations, statistical regression and articial neural networks (ANNs). In

tudy is conducted on the prediction of the bubble point pressure and oil

two hybrid of soft computing techniques; a genetically optimised neural

sevier.com/locate/cageo

Geosciences

A. Khoukhi / Computers & Geosciences 44 (2012) 109119110and it is needed to know when the gas is dissolved from oil. Bob isa function of pressure and temperature, which represents theratio of oil and dissolved gas volume under reservoir conditions tothe oil volume at standard conditions.

1.2. Correlation-based prediction

Since the pioneering work of Katz (1942), who developed vemethods for predicting crude oil shrinkage in 1942, a considerablevolume of literature has accumulated on the study of PVTcorrelations for crude oil. Various correlations were proposed bydifferent researchers from all over the world, with varyingdegrees of accuracy in terms of average error, and based on crudesamples from different oil elds (Katz, 1942; Glaso, 1980). Most ofthese correlations were developed empirically using graphical orregression methods. Several efforts were also made to develop auniversal correlation.

In Al-Marhoun (1988), a correlation was published for Bobusing 11,728 experimental values representing samples taken

Nomenclature

EOS Equation-of-stateFVF Formation volume factorPVT Pressure/Volume/TemperatureAPI American Petroleum InstituteANN Articial neural networksRBFNN Radial basis feed-forward neural networksGA Genetic AlgorithmANFIS Adaptive neuro-fuzzy inference systemGANFIS Genetic adaptive neuro-fuzzy inference systemGONN Genetically optimised neural networksEr Average percent relative errorEa Average absolute percent relative errorEmin Minimum root mean square errorEmax Maximum root mean square errorRMSE Root mean square errorSD Standard deviationR2 Correlation coefcientfrom 700 reservoirs, mostly from the Middle East and NorthAmerica. Other correlations were also developed for the Gulf ofSuez and Malaysian Crude oils by Al-Marhoun (1988), Macary andEl-Batanoney (1992), respectively. In Omar and Todd (1993),a reliability analysis was conducted for PVT correlations. InDe Ghetto et al. (1995), correlations were published for propertiesof UAE crude oils using 62 data sets from UAE reservoirs.A comprehensive review of various empirical correlations isreported in Almehaideb (1997) and Sutton (2008).

1.3. Articial neural networks based prediction

Several articial neural network (ANN) correlations have alsobeen proposed. The prediction performance was assessed throughfour main criteria. The rst is the average percent relativeerror (Er), which measures the relative deviation of the resultsof the prediction from the experimental data. The second is theaverage absolute percent relative error (Ea), which measuresthe relative absolute deviation of the prediction results from theexperimental values. The standard deviation (SD) measures thevariability of the obtained results with respect to the averagevalue. The forth criterion consists of the correlation coefcient R2,which indicates the degree of success in reducing the standarddeviation by regression analysis.In Elsharkawy (1998) a two hidden layers radial basis functionneural network model (RBFNM) was proposed to predict PVTproperties for crude oil and gas. The model predicts oil formationvolume factor, solution gas oil ratio, oil viscosity, saturated oildensity, under-saturated oil compressibility, and evolved gasgravity. The input data to the RBFNM were reservoir pressure,temperature, stock tank oil gravity and separator gas gravity.For the oil formation volume factor, the RBFNM resulted in abetter accuracy than as-to-date available PVT correlations. In thisstudy, the average percent relative error (Er) was estimated forthe training to 0.06% and 0.08% for testing. The average absolutepercent relative error (Ea) was obtained as 0.87% for training and0.53% for testing. The standard deviation (SD) was found to be1.28% for training and 0.57% for testing. The correlation coefcientR2 was determined as 99.46% for the training samples and 98.24%for testing.

In Varotsis et al. (1999), a novel approach for predicting thecomplete PVT behaviour of reservoir oil and gas condensates wasintroduced using an ANN. The method used key measurements

Res_Temp Reservoir temperature (1F)Mol_N2 Mole fraction of N2 (mol%)Mol_CO2 Mole fraction of CO2 (mol%)Mol_H2S Mole fraction of H2S (mol%)p Pressure (psi)Pb Bubble point pressure (psi)Pod Pressure at dead oil viscosity (psi)Bob Oil formation volume factor (RB/STB)Rs Solution gas/oil ratio, SCF/STB (m

3/m3)Rsb Bubble point solution gas/oil ratio, SCF/STB (m

3/m3)T Temperature (1F)V Volume (m3)ma Viscosity above bubble point (cP)mb Viscosity below bubble point (cP)mo Oil viscosity (cP)mob Bubble point/gas-saturated oil viscosity (cP)mod Dead oil viscosity (cP)gg Gas relative density (API)that can be performed rapidly, either in a lab or at the well site, asinput to the ANN. The ANN was trained by a PVT database of over650 reservoir uids originating from all parts of the world. Thetesting of the trained ANN indicated that, for all uid types, mostof the PVT properties estimates can be obtained with a very lowmean relative error of 0.52.5%, with no data set producing arelative error of more than 5%. This level of error is considered asbetter than that provided by tuned equation-of-state (EOS)models, which are still in common use for the estimation ofreservoir uid properties.

In Al-Marhoun and Osman (2002), two new models werepresented to predict Pb and Bob at the bubble-point pressure usinga Multi-layer Preceptron (MLP), trained by backpropagation withearly stopping, for Saudi Arabian crudes. Both models were basedon ANNs, and developed using 283 unpublished data setscollected from different Saudi Arabian elds. Off the 283 data sets,142 were used to train the Bob and Pb ANNs, 71 to cross-validate therelationships established during the training process and adjustthe calculated weights, and the remaining 70 were used to test themodel and evaluate its accuracy. The results showed that thedeveloped Bob model provides better predictions and higher accu-racy than previously published empirical correlations.

In Goda et al. (2003) another ANN correlation was developedto predict both Pb and Bob with the aid of two separate networks.

effect of each geologic driver on fractures, in an effort to assist a

2. Problem statement

lysareiso

incper

A. Khoukhi / Computers & Geosciences 44 (2012) 109119 111geologist or reservoir engineer in being able to identify, locallyand globally, the key geologic drivers affecting fractures. Auliaet al. (2010) used data mining for smart oileld reservoir analysis.In El-Sebakhy (2009), a support vector regression (SVR) was usedfor generating correlations for forecasting bubble point pressureusing three different published PVT databases, which was claimedto outperform empirical correlation and standard neural networkmodels.

All these studies prove that correlations based on data miningtechniques are more accurate than empirical correlations. How-ever, most of these correlations were found to be appropriate forthe specic region where the parameters were measured, but notfor other regions. Furthermore, the neural network correlationsdeveloped are often limited, and global correlations are usuallyless accurate compared to local correlations. Nevertheless, theachievements of neural networks opened the door to soft com-puting techniques to play a major role in the oil and gas industry.Adaptive neuro-fuzzy inference systems have been proposed as anew intelligence framework for both prediction and classicationbased on fuzzy clustering optimisation criterion and ranking(Jang et al., 1996).

This paper is an upgraded and extended version of two otherpapers (Khoukhi and Albukhitan, 2010, 2011; Khoukhi et al.,2011) that are concerned with estimating two PVT properties ofcrude oil systems; namely bubble point (Pb) and oil formationvolume factor (Bob). The main extension is the performance offurther experiments, including a genetically optimised neuralnetwork (GONN), and a genetic adaptive neuro-fuzzy inferencesystem (GANFIS), along with comparisons with standard ANN,The data used was a set of 160 measured points collected fromthe Middle East, where 120 points were dedicated to training, and20 for testing. The Bubble point network was constructed of twohidden layers, with ten neurons for each layer. All hidden neuronswere activated by a log sigmoid function. The four input datawere temperature, API gravity, gas oil ratio and gas relativedensity. The output neuron was designed to be activated withpure linear functions. The results showed that the network giveshigher accuracy in the prediction task than other publishedempirical correlations. The network has an average percentrelative error of 0.030704 and correlation coefcient of 0.9981.Other correlations were introduced in Labedi (1990) and Suttanand Farshad (1990) for Libyan and Gulf of Mexico crude oils.

In Osman and Al-Marhoun (2005), two new models weredeveloped to predict different brine properties. The rst modelpredicted brine density, oil formation volume factor (Bob),isothermal compressibility as a function of pressure and tem-perature and salinity. The second model was developed to predictbrine viscosity as a function of temperature and salinity alone.The models were developed using 1040 published data sets. Thesedata were divided into three groups: training, cross-validationand testing. Radial Basis Functions (RBF) and Multi-layer Precep-tron (MLP) neural networks were utilised in the study. Trend testswere performed to ensure that the developed model would followphysical laws. The results showed that the developed modelsoutperform the published correlations in terms of absoluteaverage percent relative error, correlation coefcient and stan-dard deviation.

1.4. Hybrid soft computing based techniques

Soft and mimetic computing techniques other than ANN werealso used (Ouenes, 2000; Jang et al., 1996). In Ouenes (2000),fuzzy neural networks were used to evaluate the hierarchicalANFIS and state-of-the-art regression based correlations.3. Proposed approach

3.1. Adaptive neuro-fuzzy inference systems (ANFIS)

Neuro-fuzzy inference systems are hybrid classication/fore-casting frameworks, which learn the rules and membershipperformance indices are detailed in Appendix II.The second comparison is through graphical representation of the

correlations between the actual and predicted values of Pb and Bob.The third approach is through graphical representation of

errors as a function of correlations (Al-Marhoun, 1988, 1992;Al-Marhoun and Osman, 2002). Scatter plots are shown for theabsolute percent relative error (Ea) versus the correlation coef-cient for all computational intelligence forecasting schemes andthe most common empirical correlations. Each modeling schemeis represented by a symbol; the good forecasting scheme shouldappear in the upper left corner of the graph.perstafunThe rst standard is an assessment through six error criteria;luding the average percent relative error (Er), average absolutecent relative error (Ea), minimum and maximum absolutecent error (Emin and Emax), root mean square errors (RMSE),ndard deviation (SD), and correlation coefcient (R2). Thesemework to other empirical correlations, statistical error ana-is and quality measures are performed. Three main standardsused in assessing the suggested techniques and their compar-ns with regression-based models.fra2.1. Data acquisition and pre-processing

Three distinct databases were used in this study to implementthe proposed techniques, to forecast both bubble point pressure(Pb) and oil formation volume factor (Bob) based on the same fourinput parameters; namely solution gasoil ratio (Rs), reservoirtemperature (T), oil gravity (API), and gas relative density (gg).The properties of the three datasets acquired are explained below;a statistical description of the datasets appears in Appendix I:

The rst dataset refers to work conducted by Al-Marhoun, whopublished correlation for estimating Pb and Bob for MiddleEastern oils (Al-Marhoun, 1988). The dataset consists of 160observations collected from 69 Middle Eastern reservoirs.

The second dataset were retrieved from works conducted byAl-Marhoun and Osman (2002). The dataset consists of 283observations collected from different Saudi Arabian oil eldsfor the prediction of Pb and Bob at Pb for Saudi crude oils.

The third dataset was obtained from the work conducted byOsman and Al-Marhoun (2005), this dataset contains 782observations collected from oil elds in Malaysia, the MiddleEast, Gulf of Mexico and Columbia.

2.2. Statistical quality measures

To compare the performance and accuracy of the proposedIn Section II, the problem is formulated and the datasets usedare introduced. Section III presents the proposed approach to theproblem and the implemented techniques. Section IV reports onthe simulation experiments and a comparative study. Section Vconcludes this work and offers perspectives and future trends.ctions from data. It is a network of nodes and directional

1996). Subtractive clustering is used in the problem at hand as itprovides a fast one-pass algorithm to take input-output trainingdata to generate an adaptive fuzzy inference system that is bettertailored to the dataset. The subtractive clustering methodassumes each data point is a potential cluster centre, andcalculates a measure of likelihood that each data point will denea cluster centre, based on the density of the surrounding datapoints. It starts with the normalisation of all values in the datasetto t in a hypercube unit (Jang et al., 1996). Each cluster centremay be translated into a fuzzy rule for identifying a class. Thecluster radius indicates the range of its inuence. Specifying asmall cluster radius yields many small clusters in the data, thusresulting in many rules. Conversely, choosing a large radius willlead to few rules, misrepresenting therefore the data. Many trialsconsidering different radii values have led to different predictionaccuracies. A genetic algorithm (GA) is used to ne-tune theclustering parameter radii (ra), which then leads to the geneticadaptive neuro-fuzzy inference system (GANFIS). An importantliterature had been published recently on genetic fuzzy systems(GFS). Zanganeh et al., 2006 used a hybrid genetic fuzzy inferencesystem for wave parameters prediction.

The step-by-step implementation of GANFIS is clearly denedand given as follows (Fig. 2). For the training data set, each radiushas a value between a lower bound of 0.10 and an upper bound of0.90. The tness function used for the genetic search is the leastsquared error between the predicted and target value for the

A. Khoukhi / Computers & Geosciences 44 (2012) 109119112links. Associated with the network is a learning rule, for instance,back-propagation. These networks are learning a relationshipbetween inputs and outputs. This type of network covers anumber of different approaches, namely Mamdani type andTakagiSugenoKang (TSK) type (see Jang et al., 1996) for moredetail. Unlike the Mamdani method, also referred to as subjectivefuzzy modelling as it builds the fuzzy if-then rules through expertstatements which might involve vagueness and subjectivity,the TSK fuzzy objective modelling method is a framework forgenerating fuzzy if-then rules from input/output numerical data.A way to construct a TSK fuzzy model from numerical dataproceeds in three steps: fuzzy clustering, setting of the member-ship functions, and parameter estimation (Nikravesh et al., 2003;Jang et al., 1996). Fig. 1 shows an ANFIS System with a two-inputstwo-rules one-output arrangement.

The implemented ANFIS in the study at hand is made up of sixlayers. The rst layer is the input layer, characterising the crispinputs. The second layer performs the fuzzication of the crispinputs into linguistic variables, through Gaussian transfer func-tions. The third is the rule layer, which applies the product t-normto produce the ring strengths of each rule. This is followed by anormalisation layer, at which each node calculates the ratio of arules ring strength to the sum of the ring strengths of all rules.The fth layer performs the defuzzication. The last layer con-ducts the aggregation, where an output is obtained as thesummation of all incoming signals. The training rule option usedis the LevenbergMarquard version of the gradient back-propaga-tion algorithm.

Fuzzification T-norm Normalization Defuzzification1 2x x

1 2x x

+

2 2w y

1 1w y

y

N

11

1 2

ww

w w

22

1 2

ww

w w

1w

N2w

11

12

21

22

2x

1x

=

+

+=

Fig. 1. ANFIS system with two inputs two-rule one-output.3.2. Genetic adaptive neuro-fuzzy inference system (GANFIS)

Genetic Algorithms (GAs) are intelligent search mechanismsbased on the principle of natural selection and populationgenetics that are transformed by three genetic operators: selec-tion, crossover and mutation. Each string (chromosome) is apossible solution to the problem being optimised, and each bit(or group of bits) represents a value or of some variable (gene) ofthe problem. These solutions are classied by an evaluationfunction, giving better values, or tness, for better solutionsindividuals. Each solution must be evaluated by the tnessfunction to produce a value. Different crossover and mutationrates are used for the optimisation using genetic algorithms. Theability of genetic programming and adaptive neuro-fuzzy infer-ence system (ANFIS) techniques were considered for groundwater depth forecasting (Shiri and Kis-i, 2010).

To reduce the number of rules that are generated during ANFISimplementation, different clustering methods have been pro-posed, e.g., subtractive clustering and fuzzy-C means (Jang et al.,output variable.A population initialisation of size N and a termination criterion

are rst dened. The termination is either the maximum numberof generations or errors. Then the number n of radii are generatedrandomly radii[r1,r2, y, rn] in the solution space. Note hereradiis are synonymous with chromosomes, also called indivi-duals. Therefore each population now has (Nn) chromosomes.

The whole population in the generation is evaluated throughANFIS, using the root mean squared error of the testing data as acriterion of best t. Candidate chromosomes for the new generationare selected from the population, using a tournament selectionprocedure. Then, crossover and mutation operations are performed

No1t t= +

No

i = i+1

Run Genetic operators

Run ANFIS

Yesi nN

Data Reading

RMSE<

Yes

No t > T Yes

Select clustering radiustir

Run ANFIS

Data Normalisation

t =1

Generate N individual 1[ ,.., ]t t tNr r r=

i =1

RMSE<

Yes

No

Stop the program with tr as

clustering radii Fig. 2. Flowchart of the proposed GANFIS approach.

on the selected chromosomes. The Nn best individuals are selectedfrom the parent and child populations (elitism). These new indivi-duals will form the parent population for the next generation. Theprocess is repeated until the termination criterion is met.

Other GA algorithm settings consist of the number of genera-tions (20), and population size (10). Population type (double),Selection function: (Stochastic uniform), Crossover function:(Scattered with crossover rate: 0.80), Mutation function:(Gaussian with mutation rate: 0.20). The other options of ANFISare set to Matlab default. The results for GANFIS show a sig-nicant improvement in both the bubble point pressure (Pb) andoil formation volume factor (Bob) predictions than when the trial-and-error selection method was used for the radii.

3.3. Genetically optimized neural networks (GONN)

At this stage we consider a hybrid genetic neural network forthe Pb and Bob (at Pb) prediction problem.

Recently, several hybrids of genetic algorithm with ANNs wereproposed [Balan et al., 1996]. In Oloso et al. (2009) a differentialevolutionary articial neural network was introduced for predictingviscosity and gas/oil ratio curves.

schemes. Based on the results obtained within the external

Test ANN Generate N individual 11 ,...,

NRw w

A. Khoukhi / Computers & Geosciences 44 (2012) 109119 113 Train ANN

Generate New population

Run Generate operators

No

i = i+1Yes

Select weight vector

Train ANN

i =1

Stop, Display optimal weights

i N

No

Yes

RMSE

correlations methods. In addition, these schemes have a highaccuracy in predicting Pb and Bob values with stable performancesand achieved the lowest absolute percent relative errors, lowestminimum errors, lowest maximum errors, lowest root meansquare errors, and highest correlation coefcients among theother correlations for the three distinct databases used. Never-theless, because the three datasets used are from differentlocations and reservoirs, the impact produced on the predictionperformance of each dataset is independent and not signicantlycorrelated to the other. Henceforth, the results of the presentstudy are discussed for each data set. Moreover, as noted in theintroduction, most of the empirical correlations developed, orthose developed using graphical or regression methods, are onlyapplicable to a specic region and data set, effort is still beingmade to come up with a universal correlation.

Figs. (4)(8) illustrate a sample of the scatter plots of thepredicted results versus the experimental data for Pb and Bob

values using the distinct data sets provided at training and testingstages. These cross-plots indicate the degree of agreementbetween the actual (experimental) and the predicted values. Foran ideal model, all points on the plot should appear on a line ofangle of 451 with respect to the x-axis, with equidistant pointsfrom both axes and with a correlation coefcient equal to 1.

Also as it is clear from the Figures, that the regression modelshave not good generalization capabilities as compared to softcomputing based models. This fact was also reported in a previousstudy by Al-Marhoun (1988, 1992), Al-Marhoun and Osman(2002). They noticed especially that the developed regressionmodels using Middle East crude oils data, when tested on newdata from different crudes, they fail to provide with acceptableperformance.

Fig. 9 shows a scatter plot of Ea versus R2 at the training stagefor all modeling schemes that are used to determine Bo based onthe data set used in Osman and Al-Marhoun (2005). We observe

Fig. 4. Correlation coefcient and cross-plot for GONN Bob (left) and Pb (right) prediction (1st dataset).

A. Khoukhi / Computers & Geosciences 44 (2012) 109119114Fig. 5. Correlation coefcient and cross-plot of GANFIS for Bob and Pb (1st dataset).

A. Khoukhi / Computers & Geosciences 44 (2012) 109119 115that the symbol corresponding to the GONN scheme falls in theupper left corner with Ea0.10015% and R20.9999, whileGANFIS very close to GONN with Ea0.1013% and R20.9998;whereas standalone ANN got Ea1.7886% and R20.9878.

Fig. 7. Correlation coefcient and cross-plo

Fig. 8. Bob Correlation coefcient and cross-plot of regression models. (t

Fig. 6. Pb Correlation coefcient crosEmpirical correlations indicate higher error values with lowercorrelation coefcients, such as, Standing with Ea2.1923% andR20.9874; Al-Marhoun with Ea2.1037% and R20.9846; andGlaso Correlation with Ea2.8173% and R20.97813.

ts of ANN for Bob and Pb (1st dataset).

op to bottom: (a) Al-Marhoun (b) Standing, (c) Glaso) (1st dataset).

s-plot for GANFIS (2nd dataset).

Fig. 10. Performance criteria obtained using dataset 2 for the eight methods forbubble point pressure (Pb).

0.9

0.92

0.94

0.96

0.98

1

1.02

0

Cor

rela

tion

Coe

ffici

ent R

2

Absolute Percent Relative Error EA

ANN

Standing

Marhoun

Glaso

GANFIS

GONN

0.5 1 1.5 2 2.5 3 3.5

Fig. 9. Performance of different techniques (3rd dataset Correlation coefcient vs.absolute percent relative error).

A. Khoukhi / Computers & Geosciences 44 (2012) 109119116Figs. 10 and 11 show different error criteria obtained usingdataset 2 for the six methods for bubble point pressure (Pb) andoil formation volume factor (Bob), respectively. The gures showclearly that the GANFIS and GONN are very competitive andproduce the best performance, as compared to standalone ANN

Fig. 11. Performance criteria obtained using dataset 2 for the eight methods for oilformation volume factor (Bob).

Table 1Time complexity of all models for Pb Prediction (1st dataset).

Model Training Testing

Cputime (s)

Standing 34.819 0.0682

Marhoun 15.254 0.0782

Glaso 19.025 0.072

ANN 724.54 0.088

GONN 57600 0.0182GANFIS 69200 0.0096

and common empirical correlations including the correlationsby Al-Marhoun (1988), Osman and Al-Marhoun (2005), andAl-Marhoun and Osman (2002)) and multi-layer back-propaga-tion neural networks. The six considered error criteria, include theaverage percent relative error (Er), average absolute percentrelative error (Ea), minimum and maximum absolute percenterror (Emin and Emax), root mean square errors (RMSE), standarddeviation (SD), and correlation coefcient (R2).

Finally, regarding time complexity, Table 1 shows the compu-tational processing time for all models for the Pb prediction. It isclear that GANFIS and GONN are very demanding in computa-tional time, as compared to stand alone ANN and regression basedmodels. However this does not jeopardize usage as these compu-tations are performed ofine in the development stage, and oncethe parameters of GANFIS and GONN are optimized these struc-tures are then used for online calculations with signicantlylower computational time as compared to ANN, but with betterprediction accuracy.

5. Conclusion and future work

In this paper, a comprehensive study is conducted on theprediction problem of PressureVolumeTemperature (PVT)properties, namely bubble point pressure and oil formationvolume factor. Several soft computing techniques are explored;including articial neural networks, and adaptive neuro-fuzzy

inference systems. The paper also presented two hybrids; agenetically optimised neural network and a genetically enhancedsubtractive clustering technique for parameter identication ofthe adaptive neuro-fuzzy inference system.

Three distinct published databases were utilised to investigate thecapabilities of the soft computing techniques. Based on simulations,and by comparing the results obtained, one can conclude that, thegenetic neural network and the genetic neuro-fuzzy inference systemschemes showed better performance in predicting Pb and Bob valueswith stable performance, and achieved the lowest absolute percentrelative error, lowest minimum error, lowest maximum error, lowestRMSE and the highest correlation coefcient R2 compared to themany existing empirical correlations used for the three distinct datasets. The plan is to push this hybrid computing further by incorpor-ating some recent advances in soft computing, like optimized supportvector regression and extreme learning machines and granularcomputing (Khoukhi et al., 2011; Zhu et al., 2005; Xu and Shu,2006; Scholkopf and Smola, 2002; Yu and Pedrycz, 2009) and otherintelligent search techniques. Some these hybrids are being tested inongoing works. The resulting massive computational time required toimplement these hybrids will be tackled using the High PerformanceComputing (HPC) facilities available to achieve higher performancewith a reasonable time complexity. Furthermore, the proposed hybridevolutionary soft computing techniques are exible, reliable and can

Table A.1Statistical description of dataset 1 used for PVT models (160 records).

Parameter Min Max Average SD

Input variablesTemperature (Tf), (1F) 74 240 144.43 39.036Gasoil ratio (Rs), (SCF/STB) 26 1602 557.66 403.12

Gas relative density (gg) 0.69 1.367 0.96417 0.17103Api oil gravity, (degrees api) 19.4 44.6 32.388 5.7444

Output variables

ds).

Input variables

ds).

A. Khoukhi / Computers & Geosciences 44 (2012) 109119 117Temperature (Tf ), (1F) 75Gasoil ratio (Rs) (SCF/STB) 24

Gas relative density (gg) 0.7527API oil gravity (degrees API) 17.5

Output variablesBubble point pressure (Pb, psi) 90

Oil FVF at Pb, (RB/STB) 1.0308

Table A.3Statistical description of dataset 3 used for PVT models (782 Recor

Parameter Min

Input variablesTemperature (Tf ), (1F) 58Gasoil ratio (Rs) (SCF/STB) 8.61

Gas relative density (gg) 0.511API oil gravity (degrees API) 11.4

Output variablesBubble point pressure (Pb), (psi) 107.33

Oil FVF at Pb, (RB/STB) 1.028Bubble point pressure (Pb), (psi) 130 3573 1731.1 1084.6

Oil FVF at Pb, (RB/STB) 1.032 1.997 1.3036 0.20564

Table A.2Statistical description of dataset 2 used for PVT models (283 Recor

Parameter MinAppendix I. Statistical description of datasets used

See Tables (A.1)(A.3).

Max Average SD

240 147.35 47.772

1453 432.46 303.58

1.8195 1.008 0.14826

44.6 31.622 5.2518

3331 1390.2 860.66

1.889 1.2504 0.15824

Max Average SD

341.6 181.9 51.984

3617.3 541.75 483.68

1.789 0.88825 0.18556

63.7 34.588 8.7286

7127 2006.1 1291.2

2.887 1.3362 0.27201Acknowledgments

This work is supported by King Fahd University of Petroleumand Minerals under Grant no. SB100014.be implemented for other related oil and gas industry problems,especially in the prediction of permeability and porosity, historymatching, data management and rock mechanics properties. Some ofthese tasks are being undertaken in ongoing works.

A. Khoukhi / Computers & Geosciences 44 (2012) 109119118Appendix II. Performance indices

Average percent relative error: Measures the relative devia-tion from the experimental data, given as:

Er 1n

Xni 1

Ei II:1

where Ei is a relative deviation of an estimated value from anexperimental value.

Ei BobexpBobest

Bobexp

" #i

100 i 1,2,. . .n II:2

Average absolute percent relative error: Measures the rela-tive absolute deviation from the experimental values, dened as:

Ea 1

n

Xni 1

9Ei9 II:3

Minimum absolute percent relative error: To dene therange of error for each correlation, the calculated absolute percentrelative error values are scanned to determine the minimumvalues. They are dened by:

Emin minn

i 19Ei9 II:4

Maximum absolute percent relative error: Similarly, themaximum absolute percent relative error is

defined as : Emax maxn

i 19Ei9 II:5

Root mean squares error: Measures the data dispersionaround zero deviation, dened by:

RMSE 1n

Xni 1

E2i

" #1=2II:6

The correlation coefcient: Represents the degree of successin reducing the standard deviation by regression analysis. It isdened by:

R1

Xni 1

BobexpBobest2i =Xni 1

BobexpBob2i

vuut II:7where

Bob 1

n

Xni 1

Bobexpi II:8

Appendix III. Regression models used (Standing, 1947;Glaso, 1980; Al-Marhoun, 1992)

Standing (1947) proposed the following empirical expressionsfor the formation volume factor and bubble point pressure:

Bbob 0:97590:00012 Rsgggo

0:51:25t

" #1:2III:1

Pb 18:2Rs=gg0:8310a1:4 III:2

a 0:00091T 0R 4600:0125API

Glaso (1980) suggested the following expression:

Bbob 110A III:3where

n n 2A6:585112:91329logBob0:27683logBob III:4With Bnob given as:

Bnob Rsggg0

0:5260:968T460 III:5

With respect to the bubble point pressure, he found:

logPb 1:76691:7447logPnb0:30218logBnob2 III:6

Pnb Rs=ggaTbAPIc III:7

a 0:816, b 0:172, c0:989Al-Marhoun, 1992, determined the following empirical corre-

lation of the formation volume factor with respect to the gas/oilratio, gas gravity, oil gravity and temperature:

B0 0:4970690:862963:103T0:182594:102F0:318099:105F2 III:8

with

F R2s gbggc0 and a 0:74239, b 0:323294, c120204III:9

He proposed the following for the bubble point pressure:

Pb aRbsgcggd0Tc III:10

with a5.38088103, b0.715082, c1.87784,d 3:1437 e 1:32657

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A. Khoukhi / Computers & Geosciences 44 (2012) 109119 119

Hybrid soft computing systems for reservoir PVT properties predictionIntroductionMotivation and backgroundCorrelation-based predictionArtificial neural networks based predictionHybrid soft computing based techniques

Problem statementData acquisition and pre-processingStatistical quality measures

Proposed approachAdaptive neuro-fuzzy inference systems (ANFIS)Genetic adaptive neuro-fuzzy inference system (GANFIS)Genetically optimized neural networks (GONN)

Simulation experiments and comparative studiesIntroduction and experimental set upSimulation results and comparative studies

Conclusion and future workAcknowledgmentsStatistical description of datasets usedPerformance indicesRegression models used (Standing, 1947; Glaso, 1980; Al-Marhoun, 1992)References