Recovery Rate of Vapor Extraction in Heavy Oil Reservoirs ..._M._A._et... · fractured porous media. The smart technique and statistical models describe the VAPEX ... Reservoir dip

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Recovery Rate of Vapor Extraction in Heavy OilReservoirsExperimental, Statistical, and Modeling StudiesMohammad Ali Ahmadi,† Sohrab Zendehboudi,*,‡ Alireza Bahadori,§ Lesley James,‡ Ali Lohi,∥

Ali Elkamel,⊥ and Ioannis Chatzis⊥

†Faculty of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Khuzestan, Iran‡Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada§School of Environment, Science and Engineering, Southern Cross University, Lismore, New South Wales, Australia∥Department of Chemical Engineering, Ryerson University, Toronto, Ontario, Canada⊥Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada

*S Supporting Information

ABSTRACT: The VAPor EXtraction process (also known as VAPEX) is a solvent-based enhanced oil recovery (EOR)technology that has great potential for the recovery of heavy oil and bitumen through mass transfer and gravity drainagemechanisms. In this study, laboratory tests, the multivariable regression technique, and the connectionist model optimized by aGenetic Algorithm (GA) were used to determine the oil production rate during the VAPEX process in homogeneous andfractured porous media. The smart technique and statistical models describe the VAPEX production rate in terms of threedimensionless numbers, namely the Schmidt number (Sc), the Peclet number (Pe), and a dimensionless parameter (NS) referredto as the VAPEX number. The developed smart model was constructed based on a large number of experimental data conductedunder various process conditions in both training and testing phases. A comparison of results obtained from connectionistmodeling, the regressive model, and the experimental VAPEX data exhibits an average absolute error lower than 7% between thepredicted and actual values. Using both experimental and modeling results, the statistical analysis suggests that the Peclet numberis the most important parameter affecting the oil production rate in the VAPEX, and also the smart technique is superior to theregression model developed. This study shows the effectiveness of connectionist model in prediction of VAPEX production inthe absence of sufficient laboratory and/or field data, which may lead to a proper design of heavy oil recovery schemes.

1. INTRODUCTION

The worldwide enhanced oil recovery (EOR) reviews publishedby Oil & Gas Journal during the past two decades indicate thataround a quarter of EOR production comes from nonthermalmethods used as a mean of heavy oil viscosity reduction.1−8 TheVAPor EXtraction (VAPEX) process is an emerging heavy oilrecovery technique developed first in Canada by Butler andMokrys as an analogue to the Steam Assisted Gravity Drainage(SAGD) process for heavy oil recovery.9−11 This technique takesadvantage of horizontal well technology and solvent transferfrom a gas state into bitumen by a diffusion mechanism thatcreates a low viscosity of live oil and flow conditions under theaction of gravity for recovering heavy oil and bitumen. VAPEX isassociated with injection of vapor hydrocarbon solvents, varyingfrom ethane to normal pentane, to form a vapor chamber aroundwhich the oil phase flows due to the gravity drainage mechanism.In the VAPEX process, the well configuration is the same as thatof SAGD (e.g., solvent injection takes place into the upperhorizontal well, and diluted oil along with solvent condensatesare produced from the underlying horizontal producer).9−11

The main benefits of the VAPEX process are significantlylower energy costs (both OPEX and CAPEX), potential for insitu upgrading of bitumen using solvent dilution and also itsapplicability to thin reservoirs, and reservoirs with active waterdrive or reactive mineralogy.9−24 This recovery technique couldbe employed where SAGD may fail, such as in thin reservoirs,

highly heterogeneous formations (e.g., fractured carbonates),lower oil viscosities, and lower initial oil saturation cases.14−20

Reservoir dip angle is an asset for this EOR method. AlthoughVAPEX offers a range of benefits compared to the alternativethermal EOR techniques such as SAGD and/or cyclic steamstimulation (CSS), it has two key limitations: the productionrates attained during VAPEX are noticeably lower than thoseachieved in the thermal methods, and the solvent generally costsa lot but is recoverable from the produced live oil.14−24

Some important aspects of VAPEX (e.g., productionmechanism, scale-up, screening criteria, and process modeling)are addressed in the literature.9−11,14−37 Extensive experimentaland theoretical studies along with reviews focusing on VAPEXare also reported.9−11,14−37 For example, Rezaei et al. studied theoil production mechanism and recovery factor of warm VAPEXthrough a systematic experimental investigation.25−27 The warmVAPEX tests were carried out at different temperatures andpermeabilities using Cold Lake bitumen and Lloydminster heavyoil samples. They concluded that warm VAPEX is more efficientfor low permeable porous systems. Also, with an increasing levelof superheating, the potential of in situ upgrading lowered.25−27

Received: June 19, 2014Revised: September 5, 2014Accepted: September 9, 2014Published: September 9, 2014

Article

pubs.acs.org/IECR

© 2014 American Chemical Society 16091 dx.doi.org/10.1021/ie502475t | Ind. Eng. Chem. Res. 2014, 53, 16091−16106

pubs.acs.org/IECR

It is important to note that the VAPEX experiments done by thisresearch group at the University of Waterloo, Canada, wereperformed at a constant rate of live oil production,25−27 while thetests reported in the current study were conducted at a constantinjection rate of the solvent.Nenniger et al. introduced the N-Solv process for heavy oil

recovery, which is similar to a conventional VAPEX method inprocess perspectives.22,23 Following this introduction, NennigerandDunn obtained a correlation that relates the mass flow rate oflive oil to an expression, (Kϕ/μ)0.51, in which K represents themedium permeability, ϕ is the porosity of the porous system, andμ is the viscosity of original bitumen.24 On the basis of theirmodel, viscosity is the only parameter that can change theproduction rate for the N-Solv technique in a certain reservoir.24

For the first time, James reported an enhanced oil productionrate during a VAPEX process due to solvent being condensed onthe bitumen interface.19 Considerable growth in the microscopicsweep rate of bitumen was observed while conducting a porescale visualization study on glass micromodels. An increase in theamount of asphaltene precipitated was noticed during theprocess, as well. It was concluded that improvement of pore scalemixing because of drainage of the condensed solvent along theinterface of bitumen leads to a significant increase in the bitumenproduction rate.18−20

On the basis of the Butler’s equation for the VAPEXproduction rate, Yazdani and Maini obtained a correlation topredict the VAPEX recovery rate for homogeneous porousmedia.38 They also performed some simulation runs to supportthe statistical approach employed in their study.38 However, themodel was built on the basis of limited experimental data andprocess conditions. Considering the drawbacks for that study,this paper covers a large number of experiments that have beendone on both homogeneous and fractured porous systems. Inaddition, Genetics-Algorithm Artificial Neural Network (GA-ANN) was developed to overcome uncertainties and errors thatmay exist in the empirical correlations and simulation results forVAPEX.Artificial neural networks (ANNS) or connectionist models

have been effectively applied to various problems in chemical andpetroleum engineering, such as multiphase reactions, petrophys-ical properties, PVT studies, EOR methods, and membraneseparations.39−42 ANNS generally offer a proper structure andtransparency to large and multisource experimental or real datain order to develop practical quantitative systems for prediction,optimization, and explanatory purposes. This is mainly applicablefor cases where knowledge to obtain appropriate mathematicalmodels between independent and dependent variables is notsufficient or is unclear. In addition, ANNS are better designed tocapture the nonlinear nature of a certain process than empiricaland statistical relationships. As a consequence, ANN modelshave emerged as adaptable tools for data analysis and use in thenonlinear processes usually dealt with in heavy oil EORtechniques such as VAPEX. To attain a reliable ANN model,input variables and network structure should be chosen carefully.A number of evolutionary algorithms such as Unified ParticleSwarmOptimization (UPSO), Genetic Algorithm (GA), ParticleSwarm Optimization (PSO), Pruning Algorithm (PA), Imperia-list Competitive Algorithm (ICA), Shuffled Frog LeapingAlgorithm (SFLA), Hybrid Genetic Algorithm and ParticleSwarm Optimization (HGAPSO), Stochastic Particle SwarmOptimization (SPSO), and Back Propagation (BP) are generallyemployed to determine the network structure and its connectingweights.39−44 For instance, Zendehboudi et al. predicted a gas oil

ratio (GOR) with high precision for gas condensate reservoirsunder a broad range of process and thermodynamic conditionswith the aid of the ANN optimized with the Particle SwarmOptimization (PSO) algorithm.44 Table S1 in the SupportingInformation briefly presents the comparison between threecommon optimization evolutionary techniques such as GA, ICA,and PSO.This paper presents potential applications of a multivariable

regression model and feed-forward Artificial Neural Network(ANN) optimized by a Genetic Algorithm (GA) for theprediction of oil production rate in the VAPEX process. TheGA is used to determine initial weights of the factors contributedin the ANN. In the predictive tools, the VAPEX oil production isexpressed as a function of reservoir and oil properties in the formof three dimensionless numbers including the Schmidt number(Sc), Peclet number (Pe), and a dimensionless parameter (NS)introduced by Butler andMokrys.9 The developed GA-ANN andstatistical correlation are tested using data from the experimentsconducted for the purpose of the current study and some dataavailable from the literature. The predicted outputs are comparedwith the observed VAPEX data, and further details of results arediscussed throughout this paper. Such an investigation helpsselect the proper production strategy for the VAPEXmethod andassess the applicability of the process, depending on theformation characteristics and oil properties, in a time and costefficient approach.

2. THEORY2.1. VAPEX Process. The VAPEX process was primarily

proposed by Butler and Mokrys instead of SAGD for thin heavyoil reservoirs.9−11 The conventional VAPEX process involves theinjection of light hydrocarbons into a reservoir under operationalconditions where the solvent remains as a gas phase, but close todew point conditions in terms of its thermodynamic state. Thesolvent undergoes a molecular diffusion phenomenon, leading toa dramatic reduction in the oil viscosity. The viscous oil obtainssatisfactory mobility to be produced if oil dilution with thesolvent occurs adequately. The injection and production wellsare closely drilled at the bottom of the hydrocarbon formationsuch that the injection well is placed on the top of the producingwell (Figure 1). The process is advanced by the gravitationalforce. Consequently, this production technology is essentiallyslow, but high recovery factor values could be attainable. Figure 1shows a schematic of the VAPEX process.VAPEX can be employed as a hybrid process, benefiting from

combined effects of heat and solvent dilution, called the “warmVAPEX” process, which is able to enhance the low productivityof the conventional VAPEX technique.18−20,25−27,45 VAPEX canalso be utilized as a postprimary production process. The majortechnical challenge for the VAPEX process is that it has not beensuccessfully field tested yet. Thus, field injection and productionstrategies have not yet completely been understood. Underproper operational conditions, VAPEX might apply to thinnerand deeper heavy oil reserves which are beyond the reach ofsteam processes (e.g., depths > 1200 m).9−11,45 As the VAPEXprocess takes advantage of the solvent dilution of heavy oil undergravity-stabilized conditions, the solvent capillary blockage,fingering, and channeling are minimized compared to thesolvent flooding EOR methods such as gas injection and alkaliflooding.9−11,45

2.2. Dimensionless Numbers in the VAPEX Process.Butler and Mokrys proposed the following equation todetermine the VAPEX production rate:9

Industrial & Engineering Chemistry Research Article

dx.doi.org/10.1021/ie502475t | Ind. Eng. Chem. Res. 2014, 53, 16091−1610616092

ϕ= Δq Kg S hN(1.5 )So1/2

(1)

in which q represents the production rate, K introduces theformation permeability, g is the gravitational acceleration,ϕ is theporosity, So is the oil saturation, h stands for the formationthickness, and NS presents a dimensionless number which isexpressed below:

∫ ρμ

=Δ −

NC D

CC

(1 )dS

C

Cs

ss

min

max

(2)

where D and Cs are the diffusion coefficient of solvent in the oilphase and the solvent concentration, respectively.It is clear that the production rate is strongly dependent on

physical properties of the oil and solvent and reservoircharacteristics. Therefore, some dimensionless numbers interms of the above parameters can be introduced here for theprediction of oil production rate by the VAPEX process.The dimensionless production rate (Q) can be expressed by

the following equation:

μρ

=Qq

hK g (3)

By substituting eq 1 into eq 3, eq 4 is obtained as given

ϕ μρ

=Δ⎛

⎝⎜⎞⎠⎟Q

S NhK g

1.5 So2

2

1/2

(4)

The Peclet number (Pe) and Schmidt number (Sc) are twoother dimensionless parameters that can be employed for thedevelopment of predictive tools. Pe is the ratio of mass flux byconvection divided by the molecular diffusion mass flux, while Scpresents the ratio of momentum flux to mass transfer flux. Thecorresponding formulas for these two parameters are as follows:

ρμϕ

=Δ

PeKg h

S Do (5)

μρ

=ScD (6)

Appendix A presents the corresponding equations to determinethe mixture density (ρ), mixture viscosity (μ), and effectivediffusivity (D).Having four dimensionless numbers and the interactions

among them described above, the VAPEX production rate forfractured and homogeneous porous media can be modeledthrough a multivariable regression method and artificial neuralnetwork (ANN) method.

2.3. Multiple Linear Regressions. Regression is atechnique in statistics to find the proper relationship betweenan output and input parameters. The regressive models aregenerally classified into two different categories, namely linearand nonlinear.There are a number of applications for linear and/or nonlinear

regression models. The practical uses of the regressive modelsnormally fall into two broad classes as follows:(1) The first group refers to the cases in which the main goal is

to predict or forecast a target parameter through linear ornonlinear regression models using observed data series ofdependent and independent variables. It is clear that thedeveloped model should be able to estimate the output withhigh accuracy if the values of inputs are given even without havingthe real or observed response that is achieved throughexperimentation.(2) In general, the second category defines a certain objective

which is identifying the strength of the correlation betweenresponse and independent parameters. It also helps to specifywhat subsections of the input data include redundantinformation about the objective function.In many processes (or phenomena) in engineering and science

disciplines, multiple linear regression models can be employed tofind correlations to describe the objective functions in terms ofindependent variables. In chemical and petroleum engineering, itis common to transform all variables into dimensionlessparameters as the developed correlation is applicable for otherexperimental and field conditions, resulting in effectiveimplementation of process scale-up.Considering interactions between input variables, the multi-

variable linear regression model is written as follows:46,47

β β β β β β β

β

= + + + + + +

+

y x x x x x x x x x

x x xo 1 1 2 2 3 3 4 1 2 5 1 3 6 2 3

7 1 2 3 (7)

The above equation stands for a regression model with threeregressor variables (x1 − x3) and four interaction effects. βo−β7are the regression coefficients, and y is the predicted response.It should be noted here that the dependent and/or

independent variables may appear in logarithmic, exponential,or other mathematical functions within the regressioncorrelation, depending on the physics of the process involved.The methods to obtain the regression coefficients and also

evaluate regression correlations can be found in references 46and 47. Readers are encouraged to study them for moreinformation.

2.4. Artificial Neural Networks. Artificial neural networks(ANNS), which are usually called connectionist models, arecapable of categorizing highly complex relationships when theinput−output data are available.39−41,48,49 However, they areconsidered as universal function approximators that present noassessment trail from which a result can be described. Indeed,ANN is a computation model which has been developed basedon the structure, learning capability, and processing procedure ofa biological brain.39−41,48,49 A neural network is defined as a

Figure 1. A simple schematic of the VAPEX process.7



system of processing elements called neurons which makerelationships between input, hidden, and output layers. ANNmodels are determined by a training phase. For example, thenetwork is continually presented with input and output data thatare correlated in a proper manner. Although the training processis usually time-consuming, once trained, the ANN modelestimates a response almost right away.39−41,48,49 In general,there are three main categories for training methods of the ANNmodel, including supervised, unsupervised, and hybrid trainingmachines. For the first group, a right output is offered to everyinput pattern according to the connection weight. The back-propagation algorithm is a part of supervised trainingcategory.39−44,48,49 However, unsupervised training does notneed an exact answer for each input pattern throughout thelearning process. In fact, it figures out the proper structure on thebasis of the relationships between patterns in the data. TheKohonen algorithm is considered as a vital element of this type oflearning. The last category, known as hybrid learning, is acombination of supervised and unsupervised training methodssuch that the weights are obtained using either unsupervised orsupervised training techniques.39−41,48,49

The main challenges with ANN or connectionist modelinginclude how the network size is determined, how many data arerequired for each stage of the ANN system, and, last, when thetraining phase should be stopped. More theoretical and practicalinformation on connectionist modeling (e.g., ANN) can befound in these documents/papers.39−44,48,49

To assess the network performance for prediction of the targetvariable, a number of statistical parameters such as squaredcorrelation coefficient (R2), mean squared error (MSE),minimum absolute percentage error (MIPE), and maximumabsolute percentage error (MAPE) can be utilized. The MSEdefinition is given as follows:46−49

∑ ∑= −= =

Y k T kMSE12

[ ( ) ( )]k

G

j

m

j j1 1

2

(8)

where m and G represent the number of output nodes and thenumber of training samples, respectively. Yj(k) is the predictedoutput, and Tj(k) is the real value of the output variable. Theerror of the neural network lowers if the value of MSE goes tozero.2.5. Genetic Algorithm. The genetic algorithm (GA) is a

suitable methodology that is founded on a natural selectionprocess that drives biological development. GA can be used tosolve both controlled and uncontrolled optimization prob-lems.39−41,43 GA repetitively changes a population of individualsolutions. The GA randomly chooses individuals from theparents’ population at each step and employs them to createoffspring for the next generation. The population approaches anoptimum solution during succeeding generations. The GA is ableto solve various optimization cases that are not well treatedthrough conventional optimization algorithms. In particular, thegenetic algorithm is very efficient when problems includingstochastic, discontinuous, nondifferentiable, and nonlinearobjective functions are dealt with for optimization purposes.This technique has been recently used by Ahmadi et al. whileestimating permeability for heterogeneous reservoirs.43

The procedure of a typical GA in the form of a flowchart isdepicted in Figure 2.In the GA algorithm, the fitness function, F(X), can be the

same as the objective function f(X) of an unrestrainedmaximization problem (e.g., F(X) = f(X)). A minimization

problem can be altered into a maximization problem beforeimplementation of the GA. The fitness function is normally non-negative. The common transformation function for changing anunconstrained minimization problem to a fitness function isexpressed as the following:39−41,43

=+

F Xf X

( )1

1 ( ) (9)

3. METHODS3.1. Experimental Work. In order to develop a GA-ANN

model for estimation of the VAPEX production rate, a number ofexperiments were conducted (Figure 3). The data obtained fromthese experiments (e.g., 29 data) along with 171 additional dataof VAPEX experiments carried out by a number of researchers(e.g., Azin et al., Rahnema et al.) were used in this research.30−38

The oil sample employed in this laboratory study is a heavy oil of18.5° API and with a viscosity of 694 cP at ambient temperature(25 °C). This heavy oil was taken from Sarvak reservoir in Kuh-e-Mond heavy oil field located in south of Iran. Composition of theheavy oil used in this study is presented in Table 1.In addition, the solvent for the VAPEX experiments is pure

propane. Figure 3 shows a sketch of the experimental setup. Thefractures are constructed by milling all sides of the Plexiglasstripes. The fracture media are then wrapped using a wire meshto stop the particles (e.g., glass beads) from falling down into thefracture room. Black strips within the porous medium show thepresence of fractures. The setup consists of some elements suchas a VAPEX visual model, professional camcorder, mass flowmeter, gas−oil separator, solution gas collector, and samplingcontainer. The VAPEX porous system employed in this studywas a two-dimentional rectangular model. The matrix porosityand permeability are in the ranges of 25−38% and 1−100 Darcy,respectively, for the tests performed by the authors. The fractureaperture varies between 0.5 mm and 2 mm. In addition, thephysical model’s dimension is 200−700 mm (height), 175 mm(width), and 35 mm (thickness).

Test Procedure. The porous medium is initially packed withthe glass beads. To achieve the different permeability andporosity, the porous models are made with various ranges ofparticle size and a variety of fracture networks. For instance,

Figure 2. Flowchart of the GA to show the optimization methodology.



mixing glass beads BT8 and BOL 29 with a size range of 0.02−0.21 creates a porous system with a porosity of 25% and

permeability of 30 Darcy where two vertical fractures (e.g.,fracture aperture is equal to 1 mm) are placed in the porouspackaging model (e.g., run no. 3 in Table S2, SupportingInformation). The saturation process is then carried out throughvarious stages, namely, (a) heating the physical model, (b)increasing the temperature of the oil phase to about 75 °C, and(c) injecting the oil into the packed model with a pressure ofabout 10 atm. To control the setup temperature at a constantmagnitude (e.g., 35 °C), the oil-saturated porous model is placedin the air bath. After that, the inlet and exit lines are connected tothe porous model, and the experimental setup is kept at the airbath for approximately 20 h before a test begins. At the beginningof each experimental run, the separator is pressurized throughinjecting the nitrogen gas at about 10 atm. Then, the solvent (orpropane) with a pressure of 10 atm is injected at a constant rateinto the injection point that is placed at the bottom part of thephysical model. The pressure of the setup is regulated by a backpressure regulator positioned at the exit line. The regulator isnormally set to a certain pressure which is a little smaller than thepressure of the experimental setup. Also, the pressure differencebetween injecting and producing lines (e.g., wells) is about 0.2−0.3 atm as the gas production is minimized. Moreover, there is noconsiderable pressure variation along the model height. The liveoil collected in the VAPEX process is monitored via the glass part

Figure 3. A schematic of the VAPEX experimental setup [INJ, injector; PRO, producer].

Table 1. Composition of the Heavy Oil Used in This Study



in the separator. As the oil is separated, a majority of the solutiongas will be librated because of a reduction in pressure. Then, thegas and oil phases go to an accumulator vessel and an oilcontainer, respectively. Cumulative oil production and recoveryrate are recorded at various times of the VAPEX process. Inaddition, the gas produced during each trial is measured througha gas flow meter and recorded versus time. The time duration foreach run is almost 60 h, and the setup is equipped with aprofessional camcorder to record the solvent chamber evolution.It is worth noting that the experiments were repeated two orthree times to examine the reproducibility and accuracy (and/orreliability) of the results. The relative error percentage variesbetween 1.1 and 4.3% with respect to the mean value of thereplicates, exhibiting a high degree of repeatability. Thus, theaverage results were taken into account for each particular trial.3.2. Multivariable Regression Analysis.The experimental

runs were designed to investigate the impacts of Schmidt number(Sc), Peclet number (Pe), and NS number on the recovery rate inthe VAPEX process. The changes in the dimensionless numberswere attained by varying the physical properties of the oil andsolvent and characteristics of the porous systems. In addition, theexperimental data taken from the literature cover a wide range ofvalues that generate different magnitudes for dimensionlessnumbers used in this study. On the basis of the physics ofVAPEX, the following general function is expected for therecovery rate.

=Q f Sc Pe N( , , , combination of these numbers)S (10)

If the dimensionless production rate vs three variables, namely,Sc, Pe, and NS, and the interaction effect of dependentdimensionless groups (e.g., Sc·Pe·NS) is plotted in the form ofscatter plots, multivariable regression analysis can help obtain aproper statistical correlation.The validity of the regression modeling is tested using the

square of residuals, analysis of variance (ANOVA) table, andresidual plots.46,47 The ANOVA table contains informationbased on the analysis of the standard sum of the variance squaresobtained for regression purposes (see Table 3). The table givesrelevant data for two sources of deviation, which are regressionand residuals (e.g., first column of Table 3). The variation can

come from two main sources, namely, deviation of the estimatedmagnitude value from its measured value, which is called residual,and the deviation of estimated values from its group averagemagnitude that refers to regression. The total deviation is definedas the residual deviation plus the regression deviation. For each ofthese deviations, four measures of variance (e.g., columns 2through 5 in Table 3) could be defined as follows:46,47

(i) Degrees of freedom, DF: This parameter is calculated as thenumber of correlation coefficients (N) minus the numeric valueof regression variables employed in the correlation (the secondcolumn in Table 3). Reliability of the regression model increaseswith an increase in the degree of freedom.(ii) Sum of the squares, SS: This parameter represents the total

squared deviations, which are obtained using the real data andpredicted results (the third column in Table 3). SS is a criterionto determine the variance of every regression analysis. Summingup the total residual squares and the total regression squares givesthe total SS.(iii) Mean squares, MS: The fourth column in Table 3 is

composed of the sum of squares (SS) divided by the degrees offreedom (DF), which is called mean squares.(iv) F-test: This statistical parameter is employed to compare

two different regression equations in terms of the number ofregressor variables (see the fifth column in Table 3). F-testdecides if the more complex model with greater regressorparameters is necessary for prediction purposes or if the simpler(or less complex) model is proper and offers acceptable outputs.If the value of “Fobserved” is larger than the critical F which is listedin the corresponding table, the equation with more regressionvariables exhibits higher performance and is taken into account asbeing significant. In general, 0.05 is the significance level whichoffers a 95% confidence level.46,47

Another criterion to test the correctness of a regression modelis the residual plot. The difference between the actual extent ofthe target function (y) and its estimated magnitude (y)̂ is labeledas the residual parameter (e.g., “e”). The residual variable for eachparticular point is computed as expressed below:

= − ̂e y yresidual ( ) real value ( ) estimated value ( )i i i (11)

Figure 4. Architecture of a three-layer ANN for VAPEX oil production.



A residual figure shows the residual extents of the target functionversus the values of the independent parameter. The linearcorrelation is statistically appropriate if the data are randomlyscattered in the region around the x axis in the residual plot;otherwise, a nonlinear regression equation is preferred.46,47

3.3. Artificial Neural Network with Genetic Algorithm.In general, the ANN includes three steps: training, selection, andtesting. The first part is employed to train networks, the secondone to prevent overtraining the process, and the last part to makesure that the model outputs after network selection aregeneralized well. Features influencing the number of data inANN (or connectionist model) include the problem type,complexity, possible relationships between the factors, accuracy,type of the network, user experience, and the time required forsolving.39−41,48,49

In the ANN application, data selection is commonly random.If the population of the training data set is extremely small, thenetwork will be unable to be trained adequately. Consequently,the model outputs will not be reliable. Thus, having a suitableselection can ensure legitimacy of both the training and testingphases. In addition, if a small fraction of data points for the testingstage is picked, the predicted results of the testing phase appearnot to be applicable. This is due to the fact that enough pointswith various ranges should be chosen to ensure that the networkis properly working.The number of data in the training phase needs to be

optimized in order to lower unnecessary data for cases where thedata reduction does not lessen the precision of the networkprediction. The optimal number of hidden neurons is alsodetermined to decrease the time required for the network topredict the target variable. The algorithm employed for networkoptimization is the genetic algorithm (GA), and the cost functionconsidered in the GA system normally is a statistical parameter,named the mean square error (MSE). The objective in thishybrid ANN model developed here is to minimize the costfunction via the algorithm suggested in this study. Figure 4demonstrates a schematic of the neural network proposed for thecase under study.Providing more details on GA-ANN, the interconnection

weights of ANN layers are trained using GA in this study, as thehybrid technique offers the encoding measure of ANNlayers.43,44,48,49 In sum, the main stages of combination of GAwith ANN are as follows:43,44

1. The encoding weights are defined, followed by initializingthe children.

2. The network structure is specified through identification ofinput and output variables.3. On the basis of the optimization criteria, the evaluation

function is determined.4. Crossover and mutation operations lead to new generation.5. Stage 4 is continued until the evaluation function is satisfied

by the optimum values of the weights.It has been proven that the GA-ANN is able to seek in various

directions, simultaneously, to find the optimal results. Hence, theprobability to achieve a global optimum would be considerablyincreased.43,44 Considering the advantages of the hybrid smarttechnique (GA-ANN), the convergence and permutationmatters are evaded through optimization of the ANN weightsby employing GA, resulting in a lower error percentage whileforecasting the output. Besides speeding up the testing stage, thishybrid approach requires less input information for modelingand prediction purposes.43,44 More information regarding GA-ANN is available in these references.43,44

4. RESULTS AND DISCUSSIONAcceptable determination of vital parameters (e.g., oil recoveryrate, recovery factor, and oil/solvent interface velocity) in theVAPEX production process is of great importance in oil and gasenergy sectors that deal with heavy oil reservoirs. This is becausethe accurate values of these parameters are required to efficientlydesign production facilities such as wells, pumps, pipes, andseparators in terms of size, performance, and cost. It is alsobelieved that the development of strong predictive toolsconsiderably assists petroleum engineers to make wise decisionsthroughout the oil production process in terms of engineering,practical and economic prospects. Highlighting this significance,the current study introduces an appropriate and effectivetechnique through statistical analysis and connection modelingwith the aid of a variety of experimental works. On the basis of theoutputs of this laboratory and modeling investigation, it ispossible to relate an important factor, the recovery rate ofVAPEX, to main input variables such as porous media properties,process conditions, and fluid characteristics in the form ofdimensionless groups, leading to reasonable accuracy via acomprehensive procedure. It is expected that such a systematicapproach facilitates the way to achieve an optimum VAPEXproduction rate. The developed technique can be also linkedwith the petroleum engineering software packages for modelingand optimization purposes.The first part of this study includes an experimental work on

various homogeneous and fractured physical models with

Figure 5. Position of the solvent front (shown with purple color) in the fractured model during the VAPEX process at various times [matrixpermeability, 35 D; matrix porosity, 36%] based on the experiments performed by the authors.



different characteristics for the VAPEX. In the second part of thisstudy, a statistical correlation and a smart GA-ANN method areintroduced to predict the production rate of the VAPEX. Thepredictive models are built on the experimental data. The modelsdeveloped here are valid for the VAPEX process in bothhomogeneous and fractured porous systems.4.1. Experimental Phase. The experimental runs were

conducted with various fracture and matrix permeabilities. Togeneralize the results of the regression correlation and GA-ANNmodel, a variety of experimental studies available in the literaturewere also used.30−38 These research works include a variety offracture configurations, as well. A part of the VAPEXexperimental data is given in the Supporting Information(referring to the data obtained by the authors and also extractedfrom references 30−38). It should be noted that a variety ofsolvents, including methane, propane, and butane (and/ormixture of them with different composition), have been used inthe experiments. Also, the VAPEX experiments have been run atwide ranges of temperature [20−35 °C] and pressure [2−20atm] (e.g., saturation pressure and the pressure away from thisspecific pressure). Due to considerable variations in thethermodynamic conditions and various types of solvent, thediffusion coefficient is varied from 0.00001 to 0.001 cm2/min. InTable S2, the production rate is expressed in cubic centimetersper minute per unit width of the porous medium.Collecting the experimental data from different sources, this

study covers wide ranges of fluid properties and characteristics ofvarious porous systems. The movement of the solvent chamberin a particular fractured system at different times is depicted inFigure 5. As an example, the production data of the porous media(including two fractured and one homogeneous systems) with amatrix permeability of 35 Darcy and matrix porosity of 36% arealso presented in this paper. The fractured porous media includethe fracture parts whose permeabilities are 300 and 600 Darcy,respectively. Using the production history of the physical models,the recovery factor plot of the homogeneous porous system andthe fractured media during the VAPEX process is shown inFigure 6. The comparison of these porous systems with respectto production performance implies that the presence of fractureenhances the RF as it increases the effective vertical permeability,improves cross-flow of solvent and oil phases, and supplies more

area for solvent diffusing into the bulk of heavy oil within thematrix part. It is also concluded that higher fracture permeabilityattains a greater amount of oil production due to theimprovement of interaction between the matrix and fracture,and also oil and solvent contacts. Figure 7 also depicts VAPEX

production rate against time. For all three porous systems, theproduction trend mainly includes two different circumstances,namely unsteady state and pseudosteady (and/or steady state)conditions. The oil production rate experiences unstablebehavior and reaches the highest value over the transition timeperiod; however a sharp decline in the flow rate is observed, andthen a stable production rate is attained when the processundergoes the steady state condition. According to Figure 7,increasing fracture permeability increases the production rateand consequently cumulative oil production. Providing furtherjustifications for the production trend in Figure 7, an increase inhorizontal spreading velocity is observed and the solventchamber is surrounded by the medium with high oil saturationduring the transition stage. Therefore, a rapid increase in oilproduction rate occurs. Then, the horizontal velocity decreasesover time, resulting in a reduction of VAPEX oil productionthroughout a pseudo-steady-state phase. Finally, both horizontalsolvent velocity and falling oil velocity reach the steady statecondition that leads to a constant oil production rate in the rest ofthe VAPEX operation. It is worth noting that the presence offractures in the fractured media also accelerates the solventdiffusion/movement inside the high permeable media, andconsequently the VAPEX process at the initial stage experiencesa higher production rate in the fractured systems, compared tothe conventional (homogeneous) porous models, as clearlydepicted in Figure 7.To gain a better understanding of the VAPEX process prior to

statistical investigation and connectionist modeling, the impactsof important dimensionless numbers (e.g., Sc, Pe, andNS) on theVAPEX production rate are studied. Panels a, b, and c of Figure 8illustrate dimensionless inverse production rate (1/Q) versus Sc,Pe, and NS, respectively. As is clear, increasing Sc and Pe lowersthe magnitude of VAPEX production rate due to a reduction indiffusivity and oil saturation. On the other hand, as the value ofNS increases, an increase in oil production rate is noticed. Themain reason for this trend is that increasing diffusivity and

Figure 6.VAPEX recovery factor versus time for various porous systemsusing the experimental data obtained by the authors [ Matrixpermeability: 35 D, Matrix porosity: 36%].

Figure 7. Oil production rate for unfractured and fractured porousmedia during VAPEX using the experimental data obtained by theauthors [matrix porosity and permeability are 36% and 35 Darcy,respectively].



decreasing viscosity cause an improvement in the objectivefunction introduced in this study. As seen from Figure 8, it isconcluded that Pe is the most important variable that influencesthe VAPEX recovery rate. It should be also noted here that whenthe variation of a dimensionless group on the target parameter isdiscussed, the other two dimensionless numbers are keptconstant in the parametric sensitivity analysis study.4.2. Multivariable Regression Model. In the second part

of this study, the VAPEX experimental data were used to developa regressive correlation for the prediction of oil rate, which is animportant parameter to evaluate the performance of the VAPEXprocess. The dimensionless rate was defined as the ratio of the oil

recovery rate to the production rate due to the gravity for eachspecific porous system. To conduct the statistical sensitivityanalysis, the cross plots are presented to find accuratedependency of the target function to the independent variables.The natural logarithm expression is introduced in the regressioncorrelation in order to form the objective variable in a linearstructure, no matter whether the behavior trend is increasing ordecreasing. The interaction term of variables’ importance wouldbe a combined variable, representing the multiplication of thenatural logarithm of the Schmidt number (ln Sc), the naturallogarithm of the Peclet number (ln Pe), and the natural logarithmof VAPEX number (ln NS). On the basis of this, the followingrelationship for production rate was obtained:

β β β β β

β β

= + + + +

+ +

QSc Pe N Sc

Pe Sc N Sc Pe N

1ln( ) ln( ) ln( ) ln( )

ln( ) ln( ) ln( ) ln( ) ln( ) ln( )

S

S S

o 1 2 3 4

5 6(12)

Table 2 lists the values for coefficients in eq 12. As the correlationcoefficient for parameter “ln(Pe) ln(NS)” changes between

negative and positive values (e.g., lower and upper bounds). Itmeans that it may hold zero value, implying the importance ofthis parameter on the target function can be neglected.Therefore, this expression was removed from the correlation,eq 12.It should be noted here that the porosity and permeability of

fractured porous media that are required to calculate thecorresponding dimensionless variable should be effective as thecontributions of both matrix and fracture parts are considered. Inthis regard, the effective porosity (ϕe) and permeability (Ke) aredefined as follows:50,51

ϕ ϕ ϕ ϕ ϕ= + −e m f m f (13)

ϕ= + ·K K Ke m f f (14)

Here, the fracture porosity (ϕf) is defined as the void space offracture divided by the total bulk volume of porous model. Thematrix pore volume over the bulk volume of the matrix refers tothe matrix porosity (ϕm). In addition, Km and Kf represent matrixand fracture permeabilities, respectively. It is important to notethat eq 14 is used to calculate effective permeability (Ke) in TableS2 (see Supporting Information).Table 3 presents the ANOVA table for VAPEX production

rate. Since Fobserved, which is equal to 674.81, is bigger than 2.09,which corresponds to the critical F, entire parameters taken intoaccount for the regression equation of dimensionless VAPEXrate and their impacts are of high importance.46,47 Therefore, theemployed parameters cannot be deleted in order to make thestatistical correlation shorter and/or simpler.

Figure 8. Effect of the dimensionless group on VAPEX production rate:(a) Sc, (b) Pe, and (c) NS.

Table 2. Information for the Predictive Correlation ofDimensionless VAPEX Production Rate

coefficients numeric magnitude standard error lower 95% upper 95%

βo −270.29 34.73 −495.19 −45.40β1 55.59 6.83 18.36 92.83β2 0.46 0.15 0.24 1.17β3 −22.86 2.83 −40.33 −5.40β4 −5.98 0.22 −7.99 −3.96β5 4.64 0.46 1.75 7.53β6 −0.50 0.02 −0.65 −0.34



To evaluate the validity of the regression analysis conductedfor the VAPEX production rate, two residual figures were plotted.Panels a and b of Figure 9 show the residual plots for the recovery

rate with respect to “ln(Pe)” and “ln(NS),” respectively.Moreover, the residual values versus “ln(Sc) ln(Pe)” and “ln(Sc)ln(Pe) ln(NS)” are depicted correspondingly in panels a and b ofFigure 10. Since these two residual figures do not exhibit a certaintrend of dependent target variable against the parametersprovided in the “x axis,” it can be concluded that the linearregression equation is appropriate for estimation of the VAPEXproduction rate.To keep the paper at a reasonable size, the authors put just

some of the residual plots in the main text of the paper. As theparameter “ln(Pe) ln(NS)” is not considerably affecting theVAPEX recovery factor, its residual plot is not important.Therefore, it is not required for statistical assessment. Residual

plots for “ln(Sc)” and “ln(Sc) ln(NS)” are however included inthe Supporting Information if one is interested in attainingfurther information on the statistical analysis (see Figures S1 andS2).Squared residual is also a simple way to test the accuracy of a

certain linear regression. To perform a systematic comparison,the inverse VAPEX production rate determined from theregressive model is plotted versus the real data, as depicted inFigure 11. According to Figure 11, a very good match is noticedbetween predicted values and experimental outputs as thedetermination coefficient (R2) is about 0.961. Supporting thisstatement, the values of squared residuals for the developedregressive model are also shown in Table 4. Again, thesemagnitudes clearly indicate an acceptable agreement between theexperimental data and the results predicted by the linearregression analysis. In other words, a small value of the residualsand also noticeable amounts of the squared residual (e.g., > 0.95)again imply that the proposed linear regression model worksappropriately at least for the experimental conditions engaged inthis study.

Table 3. ANOVA Table for the Dimensionless ProductionRate

source DF SS MS F

regression 6 36799.57 6133.26 674.81residual 193 1754.17 9.09total 199 38553.74

Figure 9. Residual plots for two of the single parameters involved in theVAPEX correlation: (a) residual versus ln(Pe), (b) residual versusln(NS).

Figure 10. Residual plots for combinatory effects of the variablescontributed in the VAPEX production rate: (a) residuals versus “ln(Sc)ln(Pe),” (b) residuals versus “ln(Sc) ln(Pe) ln(NS)”.



4.3. GA-ANN Model. In this study, an artificial neuralnetwork (ANN) was applied to make a system to forecast theamount of oil production rate for the VAPEX method. The mainpart of the data employed in this study is based on the studies thatuse the oil samples of one of the northern Persian Gulf oil fields(e.g., Sarvak heavy oil reservoir). Data selection is a crucial stageto attain acceptable training data, leading to improvement of theANN model’s performance. Hence, a proper training phase thatcan give an appropriate level of accuracy in both training andtesting phases is required to execute to cover all of the possibleconditions. Throughout the testing process, the network shouldbe examined by a new data set which has not been employed inthe training step. Hence, the data employed in this study weredivided into two various categories: training (156 data points)and testing (44 data points). It is important to note that the datapoints for network training were chosen by a random generator.Due to different order of magnitudes of the inputs and outputs,

data normalization was performed through the followingequation as given below:

=− +

−Xx x x

x x

( )2

( )2

max min

max min(15)

where xmax and xmin stand for the highest and lowest extents ofvariable x, correspondingly.In order to optimize the neural network model, an

evolutionary algorithm (GA) was applied in this study. Theweights of the training phase were selected as parameters of anoptimization problem. The Mean Square Error (MSE) as a costfunction was considered in the GA algorithm. Back-Propagation(BP) is a gradient descent algorithm on the error space whichmight get stuck into local minima. Therefore, this algorithmstrongly depends on primary settings (weights). This flaw can befixed by employing the evolutionary algorithms (e.g., GA andPSO) that have global searching ability. The main criterion whileconducting the GA-ANN modeling was to achieve a minimumvalue for the cost function. To design a GA-ANN model, firstevery weight in the network was assumed in the range of [−1, 1].The GA-ANN was instructed through 50 iterations, followed bya BP training method. The learning coefficient was decided to be0.7, and the momentum correction factor of 0.001 was alsoutilized for the BP training procedure.Three to eight hidden neurons were tried to achieve the best

GA-ANN model in terms of statistical analysis. Table 5 lists theperformance of the hybrid models with different hidden nodes. Itwas found that an increase in the number of hidden neurons fromthree to seven enhances the precision of the ANN modeling. Itseems that the network is not able to learn the process suitablydue to a lack of adequate degrees of freedom, when the numberof hidden neurons are decided to be three, five, and six. However,a decline in the performance of GA-ANN is noticed when thehidden neurons are eight. Hence, the optimal nodes in thehidden layer are seven in the developed smart technique (e.g.,GA-ANN), since at eight hidden neurons it takes a long time forthe GA-ANN to be trained, and the data might be overfitted.In order to assess the performance of the hybrid GA-ANN

algorithm, a back-propagation neural network (BP-ANN) wasconstructed with the same data employed in the GA-ANNmodel. A comparison between predicted and measurednormalized oil production magnitudes at training and testingstages for both hybrid BP-ANN and GA-ANN systems is shownin Figures 12 and 13. Evidently, the outputs of the modelsimulated with testing data for both ANNmodels is in reasonableagreement with the experimental VAPEX data (Figures 12 and13). However, higher accuracy is clearly observed in Figure 13 forthe GA-ANN. This means that training the neural network usingthe GA algorithm leads to better results than the BP algorithm.The simulation performance of the BP-ANN and GA-ANN

models was examined based on the efficiency coefficient (R2),mean square error (MSE), maximum absolute percentage error(MAPE), and minimum absolute percentage error (MIPE). Theparameters R2 = 0.979, MSE = 0.061, MIPE = 0.517, and MAPE= 39.438 for the hybrid smart tool compared to R2 = 0.906, MSE= 0.304, MIPE = 0.973, and MAPE = 92.721 for BP-ANNconfirm remarkable performance of GA-ANN (Table 6; Figures

Figure 11. Comparison between the experimental VAPEX oil recoveryrate and the results obtained from the statistical correlation, eq 12.

Table 4. Summary of the Statistical Linear Regression for theVAPEX Production Rate

parameter value

multiple R 0.98R square 0.96standard error 3.01observations 200

Table 5. Performance of the GA-ANN Based on the Number of Hidden Neurons

training testing

number of hidden neurons R2 MSE MIPE (%) MAPE (%) R2 MSE MIPE (%) MAPE (%)

3 0.842 0.921 0.754 54.617 0.760 0.234 2.347 65.0985 0.883 0.752 0.673 46.440 0.887 0.175 1.780 53.3396 0.934 0.503 0.534 41.246 0.920 0.097 0.823 42.8657 0.991 0.362 0.312 36.113 0.979 0.061 0.517 39.4388 0.935 0.454 0.634 40.020 0.901 0.091 0.749 41.003



13 and 14). An R2 > 0.9 normally implies a very satisfactoryperformance, while an R2 extent between 0.8−0.9 exhibits a goodperformance. Also, values less than 0.8 indicate an unacceptablemodel performance.46,47

The match between the measured and predicted oilproduction values via BP-ANN and GA-ANN models in termof scatter diagrams is demonstrated in Figures 14 and 15. TheGA-ANN obviously provides results in good agreement with theexperimental data. In the case of an excellent match between themeasured and predicted values, the estimated values lie in thediagonal line. As about all of the data fall on this particular line (Y= X), it implies the acceptable precision of the GA-ANN system.It conveys the message that the GA-ANN network is able toprevent the model from being stuck in local optima. The mainreason for this capability is that both global and local searchingcharacteristics are included in the GA-ANN because of thepresence of GA and BP at the same time.Figures 16 and 17 present the performance plots for the BP-

ANN model and the hybrid ANN system (e.g., GA-ANN),respectively. The figures illustrate how validation, best, training,and testing models launched are related to each other forestimating the VAPEX production rate, in terms ofMSE against anumber of epochs. As indicated with blue circles on Figures 16and 17, the paramount efficiency occurs at an MSE of 0.294 forthe validation phase, which corresponds to epoch 5 for the BP-ANN network, whereas the GA-ANN model undergoes the bestcondition (MSE ≈ 0.058) if the validation phase is set on 15epochs.4.4. Performance Evaluation. In this section, the perform-

ance of the proposed predictive tools is examined. The R2, MSE,

MIPE, and MAPE values for the different models are given inTable 6. Clearly, the GA-ANN offers greater performance,compared to the BP-ANN model. It is found that the utilizationof an evolutionary optimization system in the form of the GA-ANNmodel (developed in this study) leads to exceptional globaloptima and convergence rate in terms of performance. On thebasis of Table 6, it is clear that the BP-ANN shows even lowerperformance compared to the regression model. The mainreason might be the complex nature of the VAPEX process thatcauses the BP-ANN to be trapped in the local minima whilepredicting the production rate.

4.5. Relative Effects of Input Variables. In the GA-ANN,the contribution of each input parameter in the VAPEX oilproduction was obtained by a method introduced by Garson forscreening the neuronal connection weights.52 The relativeinfluence (RI) of input variables is determined if the input andoutput connection weights are known (eq 16). The higher valueof RI indicates higher correlation between the input variable andthe output variable, meaning superior importance of the variableon the amount of the target function.

Figure 12. Measured vs predicted oil recovery based on BP-ANNmodel: (a) training, (b) testing.

Figure 13. Measured vs predicted oil recovery based on GA-ANN: (a)training, (b) testing.

Table 6. Comparison of the GA-ANN System with the BP-ANN Systems in Terms of Statistical Parameters

parameters GA-ANN BP-ANN regression correlation

R2 0.979 0.906 0.961MSE 0.061 0.304 9.09MIPE 0.517 0.973 0.542MAPE 39.438 92.721 80.212



=∑

∑ ∑

= ∑

= = ∑

=

=

⎜ ⎟

⎜ ⎟

⎡⎣⎢⎛⎝

⎞⎠

⎤⎦⎥

⎡⎣⎢

⎡⎣⎢⎛⎝

⎞⎠

⎤⎦⎥⎤⎦⎥

W

WRI

jn i

i j

in

jn i

i j

1

1 1

vj

knv

kj

v vj

knv

kj

H

1

H

1 (16)

where nH and nv represent the number of hidden neurons and thenumber of input neurons, respectively. ivj refers to the absolutemagnitude of the input linking weights, and Wj introduces theabsolute linking weight between the layers associated with outputand hidden. The relative significance of Sc, Pe, and NS on the oilproduction rate is depicted in Figure 18. This figure clearly showsthat the VAPEX oil rate is affected most by Peclet number (Pe).

■ CONCLUSIONS

Employing experimental data, a multilinear regression correla-tion was presented in this paper to estimate the oil productionrate during the VAPEX process. Following the statisticalapproach, a smart technique or connectionist model includinga hybrid Genetic Algorithm and Artificial Neural Networkalgorithm (GA-ANN) was applied using the experimentalproduction history of homogeneous and fractured porousmedia. The predictive models developed relate the VAPEXrecovery rate to three dimensionless numbers such as Sc, Pe, and

NS. On the basis of the results of this study, the following mainconclusions can be drawn:1. The developed statistical and GA-ANN models are able to

predict oil production rate for the VAPEX process in bothhomogeneous and fractured systems with reasonable accuracy.

Figure 14. R2 parameter for BP-ANN model: (a) training, (b) testing.

Figure 15. R2 for GA-ANN model: (a) training, (b) testing.

Figure 16. Performance plot for the BP-ANN model.



2. The predictive performance of the proposed GA-ANNmodel is better than the conventional back-propagation ANN(BP-ANN) model and the regression correlation.3. The experimental study results show that the presence of

fracture in highly fractured media increases the VAPEXproduction rate, leading to improvement of the overall VAPEXperformance.4. GA-ANN has the potential of avoiding being stuck in local

optima in the prediction of VAPEX oil recovery as the smarttechnique employed in this research encompasses both globaland local searching capabilities.5. The predictive GA-ANN model can be combined with

heavy oil recovery modeling software available for thermalproduction methods to accelerate their efficiency, decrease theuncertainty, and amplify their forecast and modeling potentials.6. The best ANN configuration included three, seven, and one

neuron in the input, hidden, and output layers, respectively.7. The proper neural network structure was decided through a

trial and error procedure. An alternative technique is required tobe combined with the evolutionary algorithm for optimization ofthe neural network structure.8. Garson’s methodology shows that the Peclet number (Pe)

affects the VAPEX production rate the most, compared to allother dimensionless groups used in this study.

■ APPENDIX A

Mixture Density and Viscosity, and Diffusion CoefficientDensity of the oil/solvent mixture (ρ) is generally determinedusing the following equation:

ρ ρ ρ= + −C C(1 )s s s o (A.1)

where ρs and ρo are the density of solvent and oil, respectively. Csis the solvent volume fraction.In addition, the mixture viscosity is calculated based on Shu’s

correlation as follows:53

μ μ μ= −C Cs o

(1 )s s (A.2)

μs and μo represent the solvent viscosity and oil viscosity,respectively.To calculate the dimensionless numbers (e.g., Sc, Pe, and NS)

in this study, the magnitude of diffusion coefficient is required.The following correlation developed with Sigmund’s correlationis employed to obtain the effective diffusivity (D) in cm2/s:54

ρρ

α=DDo o

(A.3)

Here, ρoDo represents the density-diffusion in the dilute mixture.This property product is calculated by the equation introducedby Stewart et al.55 α is also the correction factor which is definedas follows:

α ρρ

ρρ

ρρ

= + −

+

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

0.99589 0.096016 0.25035

0.032874

c c

2

c

3

(A.4)

In eq A.4, ρc refers to the mixture critical density.

■ ASSOCIATED CONTENT*S Supporting InformationComparison between the optimization algorithms, a part of theexperimental data, and residual plots. This material is availablefree of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected], [email protected] authors declare no competing financial interest.

■ NOMENCLATURE

AcronymsANN = artificial neural networkANOVA = analysis of varianceBP = back propagationCAPEX = capital expenseCSS = cyclic steam stimulationDF = degree of freedomEOR = enhanced oil recoveryGA = genetic algorithmGOR = gas oil ratioHGAPSO = hybrid genetic algorithm and particle swarmoptimizationICA = imperialist competitive algorithmMAPE = maximum absolute percentage errorMIPE = minimum absolute percentage errorMS = mean squareMSE = mean square errorNN = neural networkOPEX = operation expensePSO = particle swarm optimization

Figure 17. Performance plot for the proposed GA-ANN model.

Figure 18. Relative effects of input parameters on the VAPEXproduction rate.



http://pubs.acs.org

mailto:[email protected]

mailto:[email protected]

RF = recovery factorRI = relative influenceSAGD = steam assisted gravity drainageSFLA = shuffled frog leaping algorithmSPSO = stochastic particle swarm optimizationSS = sum of squaresUPSO = unified particle swarm optimization

Variables

Cmax = maximum volume fraction of solvent in live oilCmin = minimum volume fraction of solvent in live oilCs = volume fraction of solvent in live oilD = diffusion coefficient of solvent in oilei = residual valueF = F valuef(x) = objective functionG = number of training samplesg = gravitational acceleration (m/s2)h = model heightivj = absolute value of input connection weightsKe = total effective permeabilityKf = fracture permeabilityKm = matrix permeabilitym = number of output nodesN = number of correlation coefficientsNS = dimensionless number defined by ButlernH = number of hidden neuronsnv = number of input neuronsPe = Peclet numberQ = dimensionless production rateq = production rateR2 = correlation coefficientSc = Schmidt numberSo = oil saturationTj(k) = actual output in ANNWj = absolute value of connection weights between hiddenand output layersxi = regression variables in eq 7y = system response in eq 7yj(k) = expected output

Greek letters

β = regression coefficients in eqs 7 and 12ϕe = total effective porosityϕf = fracture porosity, volume of the fracture over the totalbulk volume of the modelϕm = matrix porosity, pore volume of the matrix over the bulkvolume of the matrixμ = mixture viscosityμo = oil viscosityμs = solvent viscosityρ = mixture densityρo = oil densityρs = solvent density

Subscripts

e = effectivef = fracturem = matrixmax = maximummin = minimumo = oils = solvent

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Recovery Rate of Vapor Extraction in Heavy Oil Reservoirs ..._M._A._et... · fractured porous media. The smart technique and statistical models describe the VAPEX ... Reservoir dip