Recovery Rate of Vapor Extraction in Heavy Oil Reservoirs

Article pubs.acs.org/IECR

Recovery Rate of Vapor Extraction in Heavy Oil ReservoirsExperimental, Statistical, and Modeling Studies Mohammad 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 drainage mechanisms. In this study, laboratory tests, the multivariable regression technique, and the connectionist model optimized by a Genetic Algorithm (GA) were used to determine the oil production rate during the VAPEX process in homogeneous and fractured porous media. The smart technique and statistical models describe the VAPEX production rate in terms of three dimensionless numbers, namely the Schmidt number (Sc), the Peclet number (Pe), and a dimensionless parameter (NS) referred to as the VAPEX number. The developed smart model was constructed based on a large number of experimental data conducted under various process conditions in both training and testing phases. A comparison of results obtained from connectionist modeling, the regressive model, and the experimental VAPEX data exhibits an average absolute error lower than 7% between the predicted and actual values. Using both experimental and modeling results, the statistical analysis suggests that the Peclet number is the most important parameter affecting the oil production rate in the VAPEX, and also the smart technique is superior to the regression model developed. This study shows the effectiveness of connectionist model in prediction of VAPEX production in the 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 published by Oil & Gas Journal during the past two decades indicate that around a quarter of EOR production comes from nonthermal methods used as a mean of heavy oil viscosity reduction.1−8 The VAPor EXtraction (VAPEX) process is an emerging heavy oil recovery technique developed first in Canada by Butler and Mokrys as an analogue to the Steam Assisted Gravity Drainage (SAGD) process for heavy oil recovery.9−11 This technique takes advantage of horizontal well technology and solvent transfer from a gas state into bitumen by a diffusion mechanism that creates a low viscosity of live oil and flow conditions under the action of gravity for recovering heavy oil and bitumen. VAPEX is associated with injection of vapor hydrocarbon solvents, varying from ethane to normal pentane, to form a vapor chamber around which the oil phase flows due to the gravity drainage mechanism. In the VAPEX process, the well configuration is the same as that of SAGD (e.g., solvent injection takes place into the upper horizontal well, and diluted oil along with solvent condensates are produced from the underlying horizontal producer).9−11 The main benefits of the VAPEX process are significantly lower energy costs (both OPEX and CAPEX), potential for in situ upgrading of bitumen using solvent dilution and also its applicability to thin reservoirs, and reservoirs with active water drive or reactive mineralogy.9−24 This recovery technique could be employed where SAGD may fail, such as in thin reservoirs, © 2014 American Chemical Society

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. Although VAPEX offers a range of benefits compared to the alternative thermal EOR techniques such as SAGD and/or cyclic steam stimulation (CSS), it has two key limitations: the production rates attained during VAPEX are noticeably lower than those achieved in the thermal methods, and the solvent generally costs a lot but is recoverable from the produced live oil.14−24 Some important aspects of VAPEX (e.g., production mechanism, scale-up, screening criteria, and process modeling) are addressed in the literature.9−11,14−37 Extensive experimental and theoretical studies along with reviews focusing on VAPEX are also reported.9−11,14−37 For example, Rezaei et al. studied the oil production mechanism and recovery factor of warm VAPEX through a systematic experimental investigation.25−27 The warm VAPEX tests were carried out at different temperatures and permeabilities using Cold Lake bitumen and Lloydminster heavy oil samples. They concluded that warm VAPEX is more efficient for low permeable porous systems. Also, with an increasing level of superheating, the potential of in situ upgrading lowered.25−27 Received: Revised: Accepted: Published: 16091

June 19, 2014 September 5, 2014 September 9, 2014 September 9, 2014 dx.doi.org/10.1021/ie502475t | Ind. Eng. Chem. Res. 2014, 53, 16091−16106

Industrial & Engineering Chemistry Research

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ratio (GOR) with high precision for gas condensate reservoirs under a broad range of process and thermodynamic conditions with the aid of the ANN optimized with the Particle Swarm Optimization (PSO) algorithm.44 Table S1 in the Supporting Information briefly presents the comparison between three common 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 the prediction of oil production rate in the VAPEX process. The GA is used to determine initial weights of the factors contributed in the ANN. In the predictive tools, the VAPEX oil production is expressed as a function of reservoir and oil properties in the form of three dimensionless numbers including the Schmidt number (Sc), Peclet number (Pe), and a dimensionless parameter (NS) introduced by Butler and Mokrys.9 The developed GA-ANN and statistical correlation are tested using data from the experiments conducted for the purpose of the current study and some data available from the literature. The predicted outputs are compared with the observed VAPEX data, and further details of results are discussed throughout this paper. Such an investigation helps select the proper production strategy for the VAPEX method and assess the applicability of the process, depending on the formation characteristics and oil properties, in a time and cost efficient approach.

It is important to note that the VAPEX experiments done by this research group at the University of Waterloo, Canada, were performed at a constant rate of live oil production,25−27 while the tests reported in the current study were conducted at a constant injection rate of the solvent. Nenniger et al. introduced the N-Solv process for heavy oil recovery, which is similar to a conventional VAPEX method in process perspectives.22,23 Following this introduction, Nenniger and Dunn obtained a correlation that relates the mass flow rate of live oil to an expression, (Kϕ/μ)0.51, in which K represents the medium permeability, ϕ is the porosity of the porous system, and μ is the viscosity of original bitumen.24 On the basis of their model, viscosity is the only parameter that can change the production rate for the N-Solv technique in a certain reservoir.24 For the first time, James reported an enhanced oil production rate during a VAPEX process due to solvent being condensed on the bitumen interface.19 Considerable growth in the microscopic sweep rate of bitumen was observed while conducting a pore scale visualization study on glass micromodels. An increase in the amount of asphaltene precipitated was noticed during the process, as well. It was concluded that improvement of pore scale mixing because of drainage of the condensed solvent along the interface of bitumen leads to a significant increase in the bitumen production rate.18−20 On the basis of the Butler’s equation for the VAPEX production rate, Yazdani and Maini obtained a correlation to predict the VAPEX recovery rate for homogeneous porous media.38 They also performed some simulation runs to support the statistical approach employed in their study.38 However, the model was built on the basis of limited experimental data and process conditions. Considering the drawbacks for that study, this paper covers a large number of experiments that have been done on both homogeneous and fractured porous systems. In addition, Genetics-Algorithm Artificial Neural Network (GAANN) was developed to overcome uncertainties and errors that may exist in the empirical correlations and simulation results for VAPEX. Artificial neural networks (ANNS) or connectionist models have been effectively applied to various problems in chemical and petroleum engineering, such as multiphase reactions, petrophysical properties, PVT studies, EOR methods, and membrane separations.39−42 ANNS generally offer a proper structure and transparency to large and multisource experimental or real data in order to develop practical quantitative systems for prediction, optimization, and explanatory purposes. This is mainly applicable for cases where knowledge to obtain appropriate mathematical models between independent and dependent variables is not sufficient or is unclear. In addition, ANNS are better designed to capture the nonlinear nature of a certain process than empirical and statistical relationships. As a consequence, ANN models have emerged as adaptable tools for data analysis and use in the nonlinear processes usually dealt with in heavy oil EOR techniques 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 Particle Swarm Optimization (UPSO), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Pruning Algorithm (PA), Imperialist Competitive Algorithm (ICA), Shuffled Frog Leaping Algorithm (SFLA), Hybrid Genetic Algorithm and Particle Swarm Optimization (HGAPSO), Stochastic Particle Swarm Optimization (SPSO), and Back Propagation (BP) are generally employed to determine the network structure and its connecting weights.39−44 For instance, Zendehboudi et al. predicted a gas oil

2. THEORY 2.1. VAPEX Process. The VAPEX process was primarily proposed by Butler and Mokrys instead of SAGD for thin heavy oil reservoirs.9−11 The conventional VAPEX process involves the injection of light hydrocarbons into a reservoir under operational conditions where the solvent remains as a gas phase, but close to dew point conditions in terms of its thermodynamic state. The solvent undergoes a molecular diffusion phenomenon, leading to a dramatic reduction in the oil viscosity. The viscous oil obtains satisfactory mobility to be produced if oil dilution with the solvent occurs adequately. The injection and production wells are closely drilled at the bottom of the hydrocarbon formation such that the injection well is placed on the top of the producing well (Figure 1). The process is advanced by the gravitational force. Consequently, this production technology is essentially slow, but high recovery factor values could be attainable. Figure 1 shows 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 “warm VAPEX” process, which is able to enhance the low productivity of the conventional VAPEX technique.18−20,25−27,45 VAPEX can also be utilized as a postprimary production process. The major technical challenge for the VAPEX process is that it has not been successfully field tested yet. Thus, field injection and production strategies have not yet completely been understood. Under proper operational conditions, VAPEX might apply to thinner and deeper heavy oil reserves which are beyond the reach of steam processes (e.g., depths > 1200 m).9−11,45 As the VAPEX process takes advantage of the solvent dilution of heavy oil under gravity-stabilized conditions, the solvent capillary blockage, fingering, and channeling are minimized compared to the solvent flooding EOR methods such as gas injection and alkali flooding.9−11,45 2.2. Dimensionless Numbers in the VAPEX Process. Butler and Mokrys proposed the following equation to determine the VAPEX production rate:9 16092

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Appendix A presents the corresponding equations to determine the mixture density (ρ), mixture viscosity (μ), and effective diffusivity (D). Having four dimensionless numbers and the interactions among them described above, the VAPEX production rate for fractured and homogeneous porous media can be modeled through a multivariable regression method and artificial neural network (ANN) method. 2.3. Multiple Linear Regressions. Regression is a technique in statistics to find the proper relationship between an output and input parameters. The regressive models are generally classified into two different categories, namely linear and nonlinear. There are a number of applications for linear and/or nonlinear regression models. The practical uses of the regressive models normally 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 or nonlinear regression models using observed data series of dependent and independent variables. It is clear that the developed model should be able to estimate the output with high accuracy if the values of inputs are given even without having the real or observed response that is achieved through experimentation. (2) In general, the second category defines a certain objective which is identifying the strength of the correlation between response and independent parameters. It also helps to specify what subsections of the input data include redundant information about the objective function. In many processes (or phenomena) in engineering and science disciplines, multiple linear regression models can be employed to find correlations to describe the objective functions in terms of independent variables. In chemical and petroleum engineering, it is common to transform all variables into dimensionless parameters as the developed correlation is applicable for other experimental and field conditions, resulting in effective implementation of process scale-up. Considering interactions between input variables, the multivariable linear regression model is written as follows:46,47

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

q = (1.5KgϕΔSohNS)1/2

(1)

in which q represents the production rate, K introduces the formation permeability, g is the gravitational acceleration, ϕ is the porosity, So is the oil saturation, h stands for the formation thickness, and NS presents a dimensionless number which is expressed below: NS =

∫C

Cmax min

Δρ(1 − Cs)D dCs μCs

(2)

where D and Cs are the diffusion coefficient of solvent in the oil phase and the solvent concentration, respectively. It is clear that the production rate is strongly dependent on physical properties of the oil and solvent and reservoir characteristics. Therefore, some dimensionless numbers in terms of the above parameters can be introduced here for the prediction of oil production rate by the VAPEX process. The dimensionless production rate (Q) can be expressed by the following equation: qμ Q= hKρg (3)

y = βo + β1x1 + β2x 2 + β3x3 + β4 x1x 2 + β5x1x3 + β6x 2x3 + β7x1x 2x3

The above equation stands for a regression model with three regressor variables (x1 − x3) and four interaction effects. βo−β7 are 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 regression correlation, 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 46 and 47. Readers are encouraged to study them for more information. 2.4. Artificial Neural Networks. Artificial neural networks (ANNS), which are usually called connectionist models, are capable of categorizing highly complex relationships when the input−output data are available.39−41,48,49 However, they are considered as universal function approximators that present no assessment trail from which a result can be described. Indeed, ANN is a computation model which has been developed based on the structure, learning capability, and processing procedure of a biological brain.39−41,48,49 A neural network is defined as a

By substituting eq 1 into eq 3, eq 4 is obtained as given ⎛ 1.5ϕΔS N μ2 ⎞1/2 o S ⎟ Q=⎜ 2 hK ρ g ⎝ ⎠

(4)

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

Pe =

Kgρh μϕΔSoD

(5)

Sc =

μ ρD

(6)

(7)

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system of processing elements called neurons which make relationships between input, hidden, and output layers. ANN models are determined by a training phase. For example, the network is continually presented with input and output data that are correlated in a proper manner. Although the training process is usually time-consuming, once trained, the ANN model estimates a response almost right away.39−41,48,49 In general, there are three main categories for training methods of the ANN model, including supervised, unsupervised, and hybrid training machines. For the first group, a right output is offered to every input pattern according to the connection weight. The backpropagation algorithm is a part of supervised training category.39−44,48,49 However, unsupervised training does not need an exact answer for each input pattern throughout the learning process. In fact, it figures out the proper structure on the basis of the relationships between patterns in the data. The Kohonen algorithm is considered as a vital element of this type of learning. The last category, known as hybrid learning, is a combination of supervised and unsupervised training methods such that the weights are obtained using either unsupervised or supervised training techniques.39−41,48,49 The main challenges with ANN or connectionist modeling include how the network size is determined, how many data are required for each stage of the ANN system, and, last, when the training phase should be stopped. More theoretical and practical information on connectionist modeling (e.g., ANN) can be found in these documents/papers.39−44,48,49 To assess the network performance for prediction of the target variable, a number of statistical parameters such as squared correlation coefficient (R2), mean squared error (MSE), minimum absolute percentage error (MIPE), and maximum absolute percentage error (MAPE) can be utilized. The MSE definition is given as follows:46−49 MSE =

1 2

G

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

problem can be altered into a maximization problem before implementation of the GA. The fitness function is normally nonnegative. The common transformation function for changing an unconstrained minimization problem to a fitness function is expressed as the following:39−41,43 1 F (X ) = 1 + f (X ) (9)

3. METHODS 3.1. Experimental Work. In order to develop a GA-ANN model for estimation of the VAPEX production rate, a number of experiments were conducted (Figure 3). The data obtained from these experiments (e.g., 29 data) along with 171 additional data of 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 of 18.5° API and with a viscosity of 694 cP at ambient temperature (25 °C). This heavy oil was taken from Sarvak reservoir in Kuh-eMond heavy oil field located in south of Iran. Composition of the heavy 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. The fractures are constructed by milling all sides of the Plexiglas stripes. The fracture media are then wrapped using a wire mesh to stop the particles (e.g., glass beads) from falling down into the fracture room. Black strips within the porous medium show the presence of fractures. The setup consists of some elements such as a VAPEX visual model, professional camcorder, mass flow meter, gas−oil separator, solution gas collector, and sampling container. The VAPEX porous system employed in this study was a two-dimentional rectangular model. The matrix porosity and permeability are in the ranges of 25−38% and 1−100 Darcy, respectively, for the tests performed by the authors. The fracture aperture varies between 0.5 mm and 2 mm. In addition, the physical model’s dimension is 200−700 mm (height), 175 mm (width), and 35 mm (thickness). Test Procedure. The porous medium is initially packed with the glass beads. To achieve the different permeability and porosity, the porous models are made with various ranges of particle size and a variety of fracture networks. For instance,

m

∑ ∑ [Yj(k) − Tj(k)]2 k=1 j=1

(8)

where m and G represent the number of output nodes and the number of training samples, respectively. Yj(k) is the predicted output, and Tj(k) is the real value of the output variable. The error of the neural network lowers if the value of MSE goes to zero. 2.5. Genetic Algorithm. The genetic algorithm (GA) is a suitable methodology that is founded on a natural selection process that drives biological development. GA can be used to solve both controlled and uncontrolled optimization problems.39−41,43 GA repetitively changes a population of individual solutions. The GA randomly chooses individuals from the parents’ population at each step and employs them to create offspring for the next generation. The population approaches an optimum solution during succeeding generations. The GA is able to solve various optimization cases that are not well treated through conventional optimization algorithms. In particular, the genetic algorithm is very efficient when problems including stochastic, discontinuous, nondifferentiable, and nonlinear objective functions are dealt with for optimization purposes. This technique has been recently used by Ahmadi et al. while estimating permeability for heterogeneous reservoirs.43 The procedure of a typical GA in the form of a flowchart is depicted in Figure 2. In the GA algorithm, the fitness function, F(X), can be the same as the objective function f(X) of an unrestrained maximization problem (e.g., F(X) = f(X)). A minimization 16094



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

permeability of 30 Darcy where two vertical fractures (e.g., fracture aperture is equal to 1 mm) are placed in the porous packaging model (e.g., run no. 3 in Table S2, Supporting Information). The saturation process is then carried out through various 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 of about 10 atm. To control the setup temperature at a constant magnitude (e.g., 35 °C), the oil-saturated porous model is placed in the air bath. After that, the inlet and exit lines are connected to the porous model, and the experimental setup is kept at the air bath for approximately 20 h before a test begins. At the beginning of each experimental run, the separator is pressurized through injecting the nitrogen gas at about 10 atm. Then, the solvent (or propane) with a pressure of 10 atm is injected at a constant rate into the injection point that is placed at the bottom part of the physical model. The pressure of the setup is regulated by a back pressure regulator positioned at the exit line. The regulator is normally set to a certain pressure which is a little smaller than the pressure of the experimental setup. Also, the pressure difference between injecting and producing lines (e.g., wells) is about 0.2− 0.3 atm as the gas production is minimized. Moreover, there is no considerable pressure variation along the model height. The live oil collected in the VAPEX process is monitored via the glass part

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

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 16095



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

come from two main sources, namely, deviation of the estimated magnitude value from its measured value, which is called residual, and the deviation of estimated values from its group average magnitude that refers to regression. The total deviation is defined as the residual deviation plus the regression deviation. For each of these deviations, four measures of variance (e.g., columns 2 through 5 in Table 3) could be defined as follows:46,47 (i) Degrees of freedom, DF: This parameter is calculated as the number of correlation coefficients (N) minus the numeric value of regression variables employed in the correlation (the second column in Table 3). Reliability of the regression model increases with 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 and predicted results (the third column in Table 3). SS is a criterion to determine the variance of every regression analysis. Summing up the total residual squares and the total regression squares gives the total SS. (iii) Mean squares, MS: The fourth column in Table 3 is composed of the sum of squares (SS) divided by the degrees of freedom (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 of regressor variables (see the fifth column in Table 3). F-test decides if the more complex model with greater regressor parameters 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 listed in the corresponding table, the equation with more regression variables exhibits higher performance and is taken into account as being significant. In general, 0.05 is the significance level which offers a 95% confidence level.46,47 Another criterion to test the correctness of a regression model is the residual plot. The difference between the actual extent of the target function (y) and its estimated magnitude (ŷ) is labeled as the residual parameter (e.g., “e”). The residual variable for each particular point is computed as expressed below:

in the separator. As the oil is separated, a majority of the solution gas will be librated because of a reduction in pressure. Then, the gas and oil phases go to an accumulator vessel and an oil container, respectively. Cumulative oil production and recovery rate are recorded at various times of the VAPEX process. In addition, the gas produced during each trial is measured through a gas flow meter and recorded versus time. The time duration for each run is almost 60 h, and the setup is equipped with a professional camcorder to record the solvent chamber evolution. It is worth noting that the experiments were repeated two or three times to examine the reproducibility and accuracy (and/or reliability) of the results. The relative error percentage varies between 1.1 and 4.3% with respect to the mean value of the replicates, exhibiting a high degree of repeatability. Thus, the average 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 in the VAPEX process. The changes in the dimensionless numbers were attained by varying the physical properties of the oil and solvent and characteristics of the porous systems. In addition, the experimental data taken from the literature cover a wide range of values that generate different magnitudes for dimensionless numbers used in this study. On the basis of the physics of VAPEX, the following general function is expected for the recovery rate. Q = f (Sc , Pe , NS , combination of these numbers)

(10)

If the dimensionless production rate vs three variables, namely, Sc, Pe, and NS, and the interaction effect of dependent dimensionless groups (e.g., Sc·Pe·NS) is plotted in the form of scatter plots, multivariable regression analysis can help obtain a proper statistical correlation. The validity of the regression modeling is tested using the square of residuals, analysis of variance (ANOVA) table, and residual plots.46,47 The ANOVA table contains information based on the analysis of the standard sum of the variance squares obtained for regression purposes (see Table 3). The table gives relevant data for two sources of deviation, which are regression and residuals (e.g., first column of Table 3). The variation can

residual (ei) = real value (yi ) − estimated value (yi ̂ ) 16096

(11)



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

A residual figure shows the residual extents of the target function versus the values of the independent parameter. The linear correlation is statistically appropriate if the data are randomly scattered 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, and testing. The first part is employed to train networks, the second one to prevent overtraining the process, and the last part to make sure that the model outputs after network selection are generalized well. Features influencing the number of data in ANN (or connectionist model) include the problem type, complexity, possible relationships between the factors, accuracy, type of the network, user experience, and the time required for solving.39−41,48,49 In the ANN application, data selection is commonly random. If the population of the training data set is extremely small, the network will be unable to be trained adequately. Consequently, the model outputs will not be reliable. Thus, having a suitable selection can ensure legitimacy of both the training and testing phases. In addition, if a small fraction of data points for the testing stage is picked, the predicted results of the testing phase appear not to be applicable. This is due to the fact that enough points with various ranges should be chosen to ensure that the network is properly working. The number of data in the training phase needs to be optimized in order to lower unnecessary data for cases where the data reduction does not lessen the precision of the network prediction. The optimal number of hidden neurons is also determined to decrease the time required for the network to predict the target variable. The algorithm employed for network optimization is the genetic algorithm (GA), and the cost function considered in the GA system normally is a statistical parameter, named the mean square error (MSE). The objective in this hybrid ANN model developed here is to minimize the cost function via the algorithm suggested in this study. Figure 4 demonstrates a schematic of the neural network proposed for the case under study. Providing more details on GA-ANN, the interconnection weights of ANN layers are trained using GA in this study, as the hybrid technique offers the encoding measure of ANN layers.43,44,48,49 In sum, the main stages of combination of GA with ANN are as follows:43,44 1. The encoding weights are defined, followed by initializing the children.

2. The network structure is specified through identification of input 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, the probability to achieve a global optimum would be considerably increased.43,44 Considering the advantages of the hybrid smart technique (GA-ANN), the convergence and permutation matters are evaded through optimization of the ANN weights by employing GA, resulting in a lower error percentage while forecasting the output. Besides speeding up the testing stage, this hybrid approach requires less input information for modeling and prediction purposes.43,44 More information regarding GAANN is available in these references.43,44

4. RESULTS AND DISCUSSION Acceptable determination of vital parameters (e.g., oil recovery rate, recovery factor, and oil/solvent interface velocity) in the VAPEX production process is of great importance in oil and gas energy sectors that deal with heavy oil reservoirs. This is because the accurate values of these parameters are required to efficiently design production facilities such as wells, pumps, pipes, and separators in terms of size, performance, and cost. It is also believed that the development of strong predictive tools considerably assists petroleum engineers to make wise decisions throughout the oil production process in terms of engineering, practical and economic prospects. Highlighting this significance, the current study introduces an appropriate and effective technique through statistical analysis and connection modeling with the aid of a variety of experimental works. On the basis of the outputs of this laboratory and modeling investigation, it is possible to relate an important factor, the recovery rate of VAPEX, to main input variables such as porous media properties, process conditions, and fluid characteristics in the form of dimensionless groups, leading to reasonable accuracy via a comprehensive procedure. It is expected that such a systematic approach facilitates the way to achieve an optimum VAPEX production rate. The developed technique can be also linked with the petroleum engineering software packages for modeling and optimization purposes. The first part of this study includes an experimental work on various homogeneous and fractured physical models with 16097



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

different characteristics for the VAPEX. In the second part of this study, a statistical correlation and a smart GA-ANN method are introduced to predict the production rate of the VAPEX. The predictive models are built on the experimental data. The models developed here are valid for the VAPEX process in both homogeneous and fractured porous systems. 4.1. Experimental Phase. The experimental runs were conducted with various fracture and matrix permeabilities. To generalize the results of the regression correlation and GA-ANN model, a variety of experimental studies available in the literature were also used.30−38 These research works include a variety of fracture configurations, as well. A part of the VAPEX experimental data is given in the Supporting Information (referring to the data obtained by the authors and also extracted from references 30−38). It should be noted that a variety of solvents, including methane, propane, and butane (and/or mixture of them with different composition), have been used in the experiments. Also, the VAPEX experiments have been run at wide ranges of temperature [20−35 °C] and pressure [2−20 atm] (e.g., saturation pressure and the pressure away from this specific pressure). Due to considerable variations in the thermodynamic conditions and various types of solvent, the diffusion coefficient is varied from 0.00001 to 0.001 cm2/min. In Table S2, the production rate is expressed in cubic centimeters per 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 of various porous systems. The movement of the solvent chamber in a particular fractured system at different times is depicted in Figure 5. As an example, the production data of the porous media (including two fractured and one homogeneous systems) with a matrix permeability of 35 Darcy and matrix porosity of 36% are also presented in this paper. The fractured porous media include the 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 and the fractured media during the VAPEX process is shown in Figure 6. The comparison of these porous systems with respect to production performance implies that the presence of fracture enhances the RF as it increases the effective vertical permeability, improves cross-flow of solvent and oil phases, and supplies more

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

production rate against time. For all three porous systems, the production trend mainly includes two different circumstances, namely unsteady state and pseudosteady (and/or steady state) conditions. The oil production rate experiences unstable behavior and reaches the highest value over the transition time period; however a sharp decline in the flow rate is observed, and then a stable production rate is attained when the process undergoes the steady state condition. According to Figure 7, increasing fracture permeability increases the production rate and consequently cumulative oil production. Providing further justifications for the production trend in Figure 7, an increase in horizontal spreading velocity is observed and the solvent chamber is surrounded by the medium with high oil saturation during the transition stage. Therefore, a rapid increase in oil production rate occurs. Then, the horizontal velocity decreases over time, resulting in a reduction of VAPEX oil production throughout a pseudo-steady-state phase. Finally, both horizontal solvent velocity and falling oil velocity reach the steady state condition that leads to a constant oil production rate in the rest of the VAPEX operation. It is worth noting that the presence of fractures in the fractured media also accelerates the solvent diffusion/movement inside the high permeable media, and consequently the VAPEX process at the initial stage experiences a higher production rate in the fractured systems, compared to the conventional (homogeneous) porous models, as clearly depicted in Figure 7. To gain a better understanding of the VAPEX process prior to statistical investigation and connectionist modeling, the impacts of important dimensionless numbers (e.g., Sc, Pe, and NS) on the VAPEX production rate are studied. Panels a, b, and c of Figure 8 illustrate dimensionless inverse production rate (1/Q) versus Sc, Pe, and NS, respectively. As is clear, increasing Sc and Pe lowers the magnitude of VAPEX production rate due to a reduction in diffusivity and oil saturation. On the other hand, as the value of NS increases, an increase in oil production rate is noticed. The main reason for this trend is that increasing diffusivity and

Figure 6. VAPEX recovery factor versus time for various porous systems using the experimental data obtained by the authors [ Matrix permeability: 35 D, Matrix porosity: 36%]. 16098



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recovery rate to the production rate due to the gravity for each specific porous system. To conduct the statistical sensitivity analysis, the cross plots are presented to find accurate dependency of the target function to the independent variables. The natural logarithm expression is introduced in the regression correlation in order to form the objective variable in a linear structure, no matter whether the behavior trend is increasing or decreasing. The interaction term of variables’ importance would be a combined variable, representing the multiplication of the natural logarithm of the Schmidt number (ln Sc), the natural logarithm of the Peclet number (ln Pe), and the natural logarithm of VAPEX number (ln NS). On the basis of this, the following relationship for production rate was obtained: 1 = βo + β1 ln(Sc) + β2 ln(Pe) + β3 ln(NS) + β4 ln(Sc) Q ln(Pe) + β5 ln(Sc) ln(NS) + β6 ln(Sc) ln(Pe) ln(NS) (12)

Table 2 lists the values for coefficients in eq 12. As the correlation coefficient for parameter “ln(Pe) ln(NS)” changes between Table 2. Information for the Predictive Correlation of Dimensionless VAPEX Production Rate coefficients

numeric magnitude

standard error

lower 95%

upper 95%

βo β1 β2 β3 β4 β5 β6

−270.29 55.59 0.46 −22.86 −5.98 4.64 −0.50

34.73 6.83 0.15 2.83 0.22 0.46 0.02

−495.19 18.36 0.24 −40.33 −7.99 1.75 −0.65

−45.40 92.83 1.17 −5.40 −3.96 7.53 −0.34

negative and positive values (e.g., lower and upper bounds). It means that it may hold zero value, implying the importance of this 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 the corresponding dimensionless variable should be effective as the contributions of both matrix and fracture parts are considered. In this regard, the effective porosity (ϕe) and permeability (Ke) are defined as follows:50,51

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

ϕe = ϕm + ϕf − ϕmϕf

(13)

Ke = K m + ϕf ·K f

(14)

Here, the fracture porosity (ϕf) is defined as the void space of fracture divided by the total bulk volume of porous model. The matrix pore volume over the bulk volume of the matrix refers to the matrix porosity (ϕm). In addition, Km and Kf represent matrix and fracture permeabilities, respectively. It is important to note that eq 14 is used to calculate effective permeability (Ke) in Table S2 (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 into account for the regression equation of dimensionless VAPEX rate and their impacts are of high importance.46,47 Therefore, the employed parameters cannot be deleted in order to make the statistical correlation shorter and/or simpler.

decreasing viscosity cause an improvement in the objective function introduced in this study. As seen from Figure 8, it is concluded that Pe is the most important variable that influences the VAPEX recovery rate. It should be also noted here that when the variation of a dimensionless group on the target parameter is discussed, the other two dimensionless numbers are kept constant 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 develop a regressive correlation for the prediction of oil rate, which is an important parameter to evaluate the performance of the VAPEX process. The dimensionless rate was defined as the ratio of the oil 16099



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Table 3. ANOVA Table for the Dimensionless Production Rate source

DF

SS

MS

F

regression residual total

6 193 199

36799.57 1754.17 38553.74

6133.26 9.09

674.81

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

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

plots for “ln(Sc)” and “ln(Sc) ln(NS)” are however included in the Supporting Information if one is interested in attaining further information on the statistical analysis (see Figures S1 and S2). 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 the regressive model is plotted versus the real data, as depicted in Figure 11. According to Figure 11, a very good match is noticed between predicted values and experimental outputs as the determination coefficient (R2) is about 0.961. Supporting this statement, the values of squared residuals for the developed regressive model are also shown in Table 4. Again, these magnitudes clearly indicate an acceptable agreement between the experimental data and the results predicted by the linear regression analysis. In other words, a small value of the residuals and also noticeable amounts of the squared residual (e.g., > 0.95) again imply that the proposed linear regression model works appropriately at least for the experimental conditions engaged in this study.

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

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 of Figure 10. Since these two residual figures do not exhibit a certain trend of dependent target variable against the parameters provided in the “x axis,” it can be concluded that the linear regression equation is appropriate for estimation of the VAPEX production 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 the parameter “ln(Pe) ln(NS)” is not considerably affecting the VAPEX recovery factor, its residual plot is not important. Therefore, it is not required for statistical assessment. Residual 16100



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where xmax and xmin stand for the highest and lowest extents of variable x, correspondingly. In order to optimize the neural network model, an evolutionary algorithm (GA) was applied in this study. The weights of the training phase were selected as parameters of an optimization problem. The Mean Square Error (MSE) as a cost function was considered in the GA algorithm. Back-Propagation (BP) is a gradient descent algorithm on the error space which might get stuck into local minima. Therefore, this algorithm strongly depends on primary settings (weights). This flaw can be fixed by employing the evolutionary algorithms (e.g., GA and PSO) that have global searching ability. The main criterion while conducting the GA-ANN modeling was to achieve a minimum value for the cost function. To design a GA-ANN model, first every weight in the network was assumed in the range of [−1, 1]. The GA-ANN was instructed through 50 iterations, followed by a BP training method. The learning coefficient was decided to be 0.7, and the momentum correction factor of 0.001 was also utilized 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 the performance of the hybrid models with different hidden nodes. It was found that an increase in the number of hidden neurons from three to seven enhances the precision of the ANN modeling. It seems that the network is not able to learn the process suitably due to a lack of adequate degrees of freedom, when the number of hidden neurons are decided to be three, five, and six. However, a decline in the performance of GA-ANN is noticed when the hidden neurons are eight. Hence, the optimal nodes in the hidden layer are seven in the developed smart technique (e.g., GA-ANN), since at eight hidden neurons it takes a long time for the 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) was constructed with the same data employed in the GA-ANN model. A comparison between predicted and measured normalized oil production magnitudes at training and testing stages for both hybrid BP-ANN and GA-ANN systems is shown in Figures 12 and 13. Evidently, the outputs of the model simulated with testing data for both ANN models is in reasonable agreement with the experimental VAPEX data (Figures 12 and 13). However, higher accuracy is clearly observed in Figure 13 for the GA-ANN. This means that training the neural network using the 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). The parameters 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-ANN confirm remarkable performance of GA-ANN (Table 6; Figures

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

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

value

multiple R R square standard error observations

0.98 0.96 3.01 200

4.3. GA-ANN Model. In this study, an artificial neural network (ANN) was applied to make a system to forecast the amount of oil production rate for the VAPEX method. The main part of the data employed in this study is based on the studies that use the oil samples of one of the northern Persian Gulf oil fields (e.g., Sarvak heavy oil reservoir). Data selection is a crucial stage to attain acceptable training data, leading to improvement of the ANN model’s performance. Hence, a proper training phase that can give an appropriate level of accuracy in both training and testing phases is required to execute to cover all of the possible conditions. Throughout the testing process, the network should be examined by a new data set which has not been employed in the training step. Hence, the data employed in this study were divided into two various categories: training (156 data points) and testing (44 data points). It is important to note that the data points 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 following equation as given below: X=

(xmax + xmin) 2 (xmax − xmin) 2

x−

(15)

Table 5. Performance of the GA-ANN Based on the Number of Hidden Neurons training 2

number of hidden neurons

R

3 5 6 7 8

0.842 0.883 0.934 0.991 0.935

testing 2

MSE

MIPE (%)

MAPE (%)

R

0.921 0.752 0.503 0.362 0.454

0.754 0.673 0.534 0.312 0.634

54.617 46.440 41.246 36.113 40.020

0.760 0.887 0.920 0.979 0.901

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MSE

MIPE (%)

MAPE (%)

0.234 0.175 0.097 0.061 0.091

2.347 1.780 0.823 0.517 0.749

65.098 53.339 42.865 39.438 41.003



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

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

13 and 14). An R2 > 0.9 normally implies a very satisfactory performance, while an R2 extent between 0.8−0.9 exhibits a good performance. Also, values less than 0.8 indicate an unacceptable model performance.46,47 The match between the measured and predicted oil production values via BP-ANN and GA-ANN models in term of scatter diagrams is demonstrated in Figures 14 and 15. The GA-ANN obviously provides results in good agreement with the experimental data. In the case of an excellent match between the measured and predicted values, the estimated values lie in the diagonal 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 to prevent the model from being stuck in local optima. The main reason for this capability is that both global and local searching characteristics are included in the GA-ANN because of the presence of GA and BP at the same time. Figures 16 and 17 present the performance plots for the BPANN 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 for estimating the VAPEX production rate, in terms of MSE against a number of epochs. As indicated with blue circles on Figures 16 and 17, the paramount efficiency occurs at an MSE of 0.294 for the validation phase, which corresponds to epoch 5 for the BPANN network, whereas the GA-ANN model undergoes the best condition (MSE ≈ 0.058) if the validation phase is set on 15 epochs. 4.4. Performance Evaluation. In this section, the performance of the proposed predictive tools is examined. The R2, MSE,

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

GA-ANN

BP-ANN

regression correlation

R2 MSE MIPE MAPE

0.979 0.061 0.517 39.438

0.906 0.304 0.973 92.721

0.961 9.09 0.542 80.212

MIPE, and MAPE values for the different models are given in Table 6. Clearly, the GA-ANN offers greater performance, compared to the BP-ANN model. It is found that the utilization of an evolutionary optimization system in the form of the GAANN model (developed in this study) leads to exceptional global optima and convergence rate in terms of performance. On the basis of Table 6, it is clear that the BP-ANN shows even lower performance compared to the regression model. The main reason might be the complex nature of the VAPEX process that causes the BP-ANN to be trapped in the local minima while predicting the production rate. 4.5. Relative Effects of Input Variables. In the GA-ANN, the contribution of each input parameter in the VAPEX oil production was obtained by a method introduced by Garson for screening the neuronal connection weights.52 The relative influence (RI) of input variables is determined if the input and output connection weights are known (eq 16). The higher value of RI indicates higher correlation between the input variable and the output variable, meaning superior importance of the variable on the amount of the target function. 16102



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Figure 15. R2 for GA-ANN model: (a) training, (b) testing. 2

Figure 14. R parameter for BP-ANN model: (a) training, (b) testing. n ⎡⎛ ivj ⎞ ⎤ ∑ j =H 1 ⎢⎜ ∑nv i ⎟Wj ⎥ ⎣⎝ k =1 kj ⎠ ⎦ RI = ⎡ n ⎡⎛ ivj ⎞ ⎤⎤ n ∑i =v 1 ⎢∑ j =H 1 ⎢⎜ ∑nv i ⎟Wj ⎥⎥ ⎣⎝ k =1 kj ⎠ ⎦⎦ ⎣

(16)

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

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

Employing experimental data, a multilinear regression correlation was presented in this paper to estimate the oil production rate during the VAPEX process. Following the statistical approach, a smart technique or connectionist model including a hybrid Genetic Algorithm and Artificial Neural Network algorithm (GA-ANN) was applied using the experimental production history of homogeneous and fractured porous media. The predictive models developed relate the VAPEX recovery rate to three dimensionless numbers such as Sc, Pe, and

NS. On the basis of the results of this study, the following main conclusions can be drawn: 1. The developed statistical and GA-ANN models are able to predict oil production rate for the VAPEX process in both homogeneous and fractured systems with reasonable accuracy. 16103



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where ρs and ρo are the density of solvent and oil, respectively. Cs is the solvent volume fraction. In addition, the mixture viscosity is calculated based on Shu’s correlation as follows:53 μ = μsCs μo(1 − Cs)

(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 correlation is employed to obtain the effective diffusivity (D) in cm2/s:54 ρo Do α ρ

D=

(A.3)

Here, ρ D represents the density-diffusion in the dilute mixture. This property product is calculated by the equation introduced by Stewart et al.55 α is also the correction factor which is defined as follows:

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

o

o

⎛ ρ⎞ ⎛ ρ ⎞2 α = 0.99589 + 0.096016⎜⎜ ⎟⎟ − 0.25035⎜⎜ ⎟⎟ ⎝ ρc ⎠ ⎝ ρc ⎠ ⎛ ρ ⎞3 + 0.032874⎜⎜ ⎟⎟ ⎝ ρc ⎠

(A.4)

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

■

Figure 18. Relative effects of input parameters on the VAPEX production rate.

Comparison between the optimization algorithms, a part of the experimental data, and residual plots. This material is available free of charge via the Internet at http://pubs.acs.org.

2. The predictive performance of the proposed GA-ANN model 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 VAPEX production rate, leading to improvement of the overall VAPEX performance. 4. GA-ANN has the potential of avoiding being stuck in local optima in the prediction of VAPEX oil recovery as the smart technique employed in this research encompasses both global and local searching capabilities. 5. The predictive GA-ANN model can be combined with heavy oil recovery modeling software available for thermal production methods to accelerate their efficiency, decrease the uncertainty, 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 to be combined with the evolutionary algorithm for optimization of the neural network structure. 8. Garson’s methodology shows that the Peclet number (Pe) affects the VAPEX production rate the most, compared to all other dimensionless groups used in this study.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE

Acronyms

ANN = artificial neural network ANOVA = analysis of variance BP = back propagation CAPEX = capital expense CSS = cyclic steam stimulation DF = degree of freedom EOR = enhanced oil recovery GA = genetic algorithm GOR = gas oil ratio HGAPSO = hybrid genetic algorithm and particle swarm optimization ICA = imperialist competitive algorithm MAPE = maximum absolute percentage error MIPE = minimum absolute percentage error MS = mean square MSE = mean square error NN = neural network OPEX = operation expense PSO = particle swarm optimization

APPENDIX A

Mixture Density and Viscosity, and Diffusion Coefficient

Density of the oil/solvent mixture (ρ) is generally determined using the following equation: ρ = Csρs + (1 − Cs)ρo

ASSOCIATED CONTENT

S Supporting Information *

(A.1) 16104



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RF = recovery factor RI = relative influence SAGD = steam assisted gravity drainage SFLA = shuffled frog leaping algorithm SPSO = stochastic particle swarm optimization SS = sum of squares UPSO = unified particle swarm optimization

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REFERENCES

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Variables

Cmax = maximum volume fraction of solvent in live oil Cmin = minimum volume fraction of solvent in live oil Cs = volume fraction of solvent in live oil D = diffusion coefficient of solvent in oil ei = residual value F = F value f(x) = objective function G = number of training samples g = gravitational acceleration (m/s2) h = model height ivj = absolute value of input connection weights Ke = total effective permeability Kf = fracture permeability Km = matrix permeability m = number of output nodes N = number of correlation coefficients NS = dimensionless number defined by Butler nH = number of hidden neurons nv = number of input neurons Pe = Peclet number Q = dimensionless production rate q = production rate R2 = correlation coefficient Sc = Schmidt number So = oil saturation Tj(k) = actual output in ANN Wj = absolute value of connection weights between hidden and output layers xi = regression variables in eq 7 y = system response in eq 7 yj(k) = expected output Greek letters

β = regression coefficients in eqs 7 and 12 ϕe = total effective porosity ϕf = fracture porosity, volume of the fracture over the total bulk volume of the model ϕm = matrix porosity, pore volume of the matrix over the bulk volume of the matrix μ = mixture viscosity μo = oil viscosity μs = solvent viscosity ρ = mixture density ρo = oil density ρs = solvent density Subscripts

e = effective f = fracture m = matrix max = maximum min = minimum o = oil s = solvent 16105



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Recovery Rate of Vapor Extraction in Heavy Oil Reservoirs

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