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Connectionist Model to Estimate Performance of Steam-Assisted Gravity Drainage in Fractured and Unfractured Petroleum Reservoirs: Enhanced Oil Recovery Implications Sohrab Zendehboudi,*,† Amin Reza Rajabzadeh,† Alireza Bahadori,‡ Ioannis Chatzis,† Maurice B. Dusseault,§ Ali Elkamel,† Ali Lohi,∥ and Michael Fowler† †

Department of Chemical Engineering and §Department of Earth and Environmental Sciences, University of Waterloo, Waterloo, ON N2L 3G1, Canada ‡ School of Environment, Science and Engineering, Southern Cross University, Lismore, NSW 2480, Australia ∥ Department of Chemical Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada S Supporting Information *

ABSTRACT: Steam-assisted gravity drainage (SAGD) is an enhanced oil recovery technology for heavy (or viscous) oil and bitumen that involves drilling two horizontal wells in underground formations. Laboratory work, pilot-plant studies, and mathematical model development, which are generally costly, difficult, and time-consuming tasks, are taken into account as important stages in finding an effective and economical method and also predicting the performance of the SAGD technique for a certain heavy-oil reservoir. Currently, smart techniques as accurate and fairly fast tools are highly recommended for these purposes. In this work, an experimental study and an artificial neural network (ANN) linked to an optimization technique, called particle swarm optimization (PSO), were employed to obtain performance parameters such as the cumulative steam-to-oil ratio (CSOR) and recovery factor (RF) for the SAGD process. The outputs of the developed connectionist modeling (i.e., ANN− PSO) were compared with actual data, showing an average error lower than 7%, mostly because of the supremacy of the ANN− PSO method compared to the conventional ANN method and the correlations developed in this study. Furthermore, it is concluded that, among the contributing parameters, reservoir thickness and oil saturation have the most significant impacts on RF and CSOR during SAGD operations. The current study confirms the potential of hybrid connectionist modeling to screen heavy-oil fractured reservoirs for the SAGD process.

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Foster Creek facility of Cenovus Energy (as the first project) in 2001 employed the SAGD process to produce oil from a viscous oil reservoir.2−9 The economics of the SAGD process is more dependent on the steam-to-oil ratio (SOR) than on the oil production rate.10,11 Operation of SAGD at low pressures decreases the costs of generating steam. Even using inexpensive fuel for steam generation, more than 50% of the total costs is due to the steam supply, which consists of several stages such as water handling and steam transportation.10−12 It has been demonstrated that the SAGD process should be financially viable if the instantaneous SOR does not exceed 5.12−15 Therefore, when the SOR becomes too high, it is not economical to continue steam injection into the reservoir, and other production technologies should be implemented.12−15 Extremely high energy input, elevated magnitude of CO2 emission and high cost of wastewater treatment are considered as the main disadvantages of SAGD, based on the work of Deng.16 McCormack also listed some of difficulties encountered in the real cases, including (a) practical challenges with the installation of liners for horizontal wells, (b) low production rates when

INTRODUCTION Currently, the exploration and production of light or conventional oil appear to be relatively difficult and expensive tasks for the petroleum sector. In light of the difficulties, researchers and the oil-industry sector have begun to pay a great deal of attention to the development of unconventional reservoirs (e.g., those with very viscous oil) to secure energy resources for the future. Canada, with estimated oil in place (OIP) reserves of about 175 billion barrels, ranks third in the world in terms of oil reserves. Cold Lake, Athabasca, and Peace River are three regions in which oil sands are found.1−3 About 80% of heavy oil and bitumen cannot be recovered through conventional enhanced oil recovery (EOR) and mining technologies, implying that utilization of emerging in situ techniques is inevitable.2−4 Therefore, various technologies that involve injecting hot fluids and solvents have been proposed to lower the oil viscosity and also provide mechanisms for oil recovery enhancement. One common thermal method successfully implemented is steam-assisted gravity drainage (SAGD). In 1978, SAGD was invented by Imperial Oil, when it drilled the first modern horizontal well for production paired with a horizontal well for steam injection in Cold Lake, Alberta, Canada.3−7 This method was patented in 1982 by Dr. Roger Butler of Imperial Oil. After field tests performed by the Alberta Oil Sands Technology Research Authority (AOSTRA) demonstrated production success in the 1990s, commercialization of SAGD began in 2000 such that the © 2013 American Chemical Society

Received: Revised: Accepted: Published: 1645

November 11, 2012 November 22, 2013 December 2, 2013 December 2, 2013 dx.doi.org/10.1021/ie303106z | Ind. Eng. Chem. Res. 2014, 53, 1645−1662

Industrial & Engineering Chemistry Research

Article

produced using a certain EOR method. Kashihara et al. developed a stochastic approach linked to log, core, and seismic data measured at the Athabasca oil sands.26 To predict SAGD performance, the data were employed for the reconstruction of geological heterogeneities in the modeling study.26 Simulations of thermal reservoirs were also carried out by Uwiera-Gartner et al. to predict the 10-year production history of the Athabasca oil sands in Canada, which consist of shallow bitumen resources.27 Their study attempted to provide new strategies for the future development of the reservoir. Considering all advantages, modeling of the SAGD process is costly, particularly when a fine spatial grid and small time elements are employed to solve the corresponding equations in the model. On the other hand, important stages involved in reservoir modeling and optimization, such as uncertainty quantification and history matching, are generally costly, lengthy, and challenging for petroleum engineers and researchers. Such expenses and technical issues create major concerns for the proper management of SAGD operations. It is obvious that mathematical modeling of the SAGD process is a difficult and prolonged technique that requires proper theoretical knowledge, practical experience, and comprehensive experiments over various steps such as deriving the governing equations, solving the equations, and validating the results. On the other hand, some important aspects (e.g., production mechanisms and transport phenomena) might not be fully captured because of available geological and process uncertainties such that one cannot predict RF and CSOR values for SAGD technology with acceptable precision. Having addressed the main concerns with the SAGD simulations, this article presents a hybrid artificial neural network (ANN) model (specifically, ANN−PSO, where PSO is particle swarm optimization) as an appropriate predictive alternative to determine SAGD production performance for a variety of reservoirs that might contain fractures. The smart technique introduced in this study has been applied by some researchers to estimate different processes and thermodynamic characteristics in chemical, environmental, and petroleum engineering.33−41 For instance, Shafiei et al. predicted the performance of steam flooding operation in naturally carbonated fractured reservoirs using a connectionist model.33 Adams et al. used the ANN technique to develop a geochemistry−viscosity prediction method for heavy oil.34 Moreover, Abass and Song described the utilization of smart techniques to aid in the selection of EOR methods for petroleum reservoirs containing oil with different properties.35 To our knowledge, there are only a few studies in the open literature that deal with applications of connectionist modeling to heavy-oil production technologies such as vapor extraction (VAPEX), SAGD, and steam flooding.33−37 In the present research, data obtained from laboratory works and modeling simulations reported in the literature were employed to build training and testing data sets in which reservoir and steam properties were considered as the input data. Also, the training and testing stages contained the corresponding CSOR and RF as the output parameters. Required adjustments in the ANN model were also suggested to recognize and decrease extrapolations in target parameter guesses. As an important stage, statistical analysis was conducted to optimize the ANN system parameters and examine the closeness between the predicted and measured outputs. Given the reasonable results obtained, the current study using a large amount of real data offers a proficient technique for

SAGD is employed in poor and average sandstone reservoirs, (c) technical and practical restrictions on fluid removal, (d) wellbore scaling, and (e) sand production.17 These difficulties convey the message that both simulation of the production process and accurate prediction of the recovery factor and steam-to-oil ratio are necessary before the implementation of a SAGD project. Several studies discussing various aspects of thermal EOR methods (e.g., SAGD and steam flooding) can be found in the open literature.9,11,12,18−32 For example, Butler et al. (1981) developed the earliest analytical model, known as the Butler theory, for SAGD technology.18 All other models proposed afterward by researchers are extensions of the Butler model, such that they are similar in terms of fundamental formulations and mathematical methodology. However, it is worth noting that new models are capable of taking into account various aspects of SAGD, despite particular constraints. Thus, they can be used to understand SAGD physics, conduct parametric sensitivity analysis, and carry out scaleup studies. The impacts of reservoir properties such as pay thickness and porosity on the oil recovery rate in the SAGD process were also studied by Ito and Suzuki.19 According to their findings, the production rate is strongly dependent on oil saturation, pay thickness, porosity, and reservoir permeability. A new technique was introduced by Llaguno et al. in 2002 to examine the suitability of the SAGD technology for heavy-oil reservoirs in Venezuela.20 Their study resulted in an analytical tool that can be used to assess the effects of the reservoir and fluid characteristics (e.g., porosity, permeability, thickness, reservoir pressure, API gravity, oil viscosity, and oil saturation) on the recovery factor (RF) and cumulative steam-to-oil ratio (CSOR) in SAGD.20 It was found that the main parameters affecting production rate include oil saturation, porosity and, pay thickness, whereas the SOR is a function of all of the parameters.20 According to the numerical simulation work of Shin and Polikar, it was concluded that vertical permeability and reservoir thickness are two vital properties that play important roles in the performance of the SAGD process.21 In addition, their research revealed that the reservoir productivity parameter (i.e., the product of permeability times thickness, K × H) can be employed as an important screening criterion for SAGD processes.21 Using numerical modeling, Kamath et al. performed a parametric sensitivity analysis to investigate the effects of various parameters such as steam temperature, well spacing, permeability, porosity, steam quality, and shale barriers on the recovery factor and steam/oil ratio for the SAGD process implemented in heterogeneous porous media.22 A similar study was carried out by Kisman and Yeung, who used a two-dimentional numerical model, taking into account the impacts of some important parameters (e.g., relative permeability, oil viscosity, well position, flow barrier, and solution gas).23 The main difficulties with both research works are that optimum operating conditions are not addressed, and they are able to predict SAGD performance for only the particular cases studied. Recent developments in simulation software packages for chemical and petroleum engineering have enabled significant improvements in the investigation of SAGD processes. Currently, reservoir engineering simulation software packages (e.g., Eclipse and CMG) are being used as a standard tool for generating the production history of petroleum reservoirs under EOR processes such as SAGD. Simulation models are commonly used to screen oil fields. During simulation runs, a matching stage is performed based on reservoir data, and the model is run to predict the volumes (or saturation) of gas, oil, and water 1646

dx.doi.org/10.1021/ie303106z | Ind. Eng. Chem. Res. 2014, 53, 1645−1662


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Figure 1. (a) Schematic of the SAGD process. (b) Steam and oil movement inside the reservoir during the course of SAGD.42

rather than whole sections of reservoirs, to prevent steam coning. In addition, the loss to the overburden is improved as the steam chamber tends to creep below the reservoir cap.44 Condensation of the rising steam occurs on the chamber boundaries, resulting in heating of the heavy oil and, consequently, oil flow toward the production well under the force of gravity.30−32,43−45 Once the fluid communication between the injecting and producing wells is maintained for extra heavy oil, an incessant countercurrent flow including oil emulsions flowing downward and steam rising will appear quickly that causes production of the immobile oil.30−32,43−45 If a liquid pool is retained such that it bounds the producing well, the SAGD operation will be more effective in terms of thermal efficiency, avoiding steam escape from the steam chamber. This strategy is usually called steam-trap control. Through control of the temperature difference between the produced fluids and the injected steam, it is possible to monitor the presence of the liquid pool in real cases. In general, the temperature difference, known as the interwell subcool, should be in the range of 20−40 °C. Because the reservoir is usually nonuniform in terms of petrophysical characteristics, steam-trap

the accurate prediction and proper optimization of the SAGD process in a time-efficient method.

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SAGD TECHNIQUE The steam-assisted gravity drainage (SAGD) technique, introduced by Butler and co-workers (1981),18 is a countercurrent-flow process in which the driving force for steam to rise and oil to drain by gravity forces is due to the difference between the densities of oil and steam.1,7,10,29 In this process, two horizontal wells are drilled such that one is on top of the other close to the bottom of the reservoir with a distance of about 5 m from each other (see Figure 1).42 The pair of wells includes an injector and a producer. The wells drilled into oil sands typically have diameters in the range of 0.200−0.475 m and lengths of 1000−1600 m.1,7,10,43 The horizontal well placed at the top is employed to provide steam to the steam chamber that develops over it. The bottom well acts as a collection space for the fluids produced during the operation (e.g., oil emulsions, formation water, and steam condensate). According to Butler and Yee, only the reservoir part near the producing well should be heated, 1647



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Figure 2. Process flow diagram of the SAGD experimental setup. T, thermocouple; P, pressure gauge; PM, porous medium; BPR, back-pressure regulator; Inj, injector; Pro, producer.

aperture varied between 0.5 and 2 mm. The injecting and producing wells were made of stainless steel tubes with a diameter of 10 mm. The wells were perforated with a large number of ∼3-mm-diameter holes and wrapped with a mesh screen to prevent glass beads from entering the wells. The distance between the injector and producers was set at 6 cm. The matrix porosity and permeability were in the ranges of 33−38% and 20−100 Darcy, respectively. The oil sample used in this experimental investigation was a heavy oil with an API gravity and viscosity at ambient temperature (25 °C) of 18.5° and 694 cP, respectively. The heavy-oil sample used was from the Sarvak reservoir in the Kuh-e-Mond heavy-oil field, located in southern Iran. Table 1 lists the composition and additional properties of the heavy oil used in this work. Procedure. The heavy oil was homogeneously mixed with glass beads, and the slurry was carefully packed into the physical model. After the packing stage, the saturated porous medium was inserted into the insulation jacket and placed to stand vertically. Next, the model was heated to ∼45 °C to simulate the initial reservoir temperature. After that, injection of superheated steam (P = 3.8 atm and T = 160 °C) into the porous system was started. A considerable temperature difference was observed between the production and injection wells, whereas there was just a minor pressure drop between the wells. A back-pressure regulator was employed at the separator stream to control the producer pressure. The back-pressure was normally set 0.2 atm lower than the injection pressure. The flow rates (and cumulative amounts) of produced oil and water were continuously recorded

control might not be possible at one or more positions along the producing and injecting wells where the potential for loss is fairly high.30−32,43−45 Through a systematic analysis, Mohammadzadeh and colleagues accomplished a series of flow visualization tests of SAGD operation using micromodels to determine the pore-level physics of the SAGD technique both qualitatively and quantitatively, as current commercial simulator packages are not able to capture pore-level mechanisms of the process (e.g., heat-transfer, mass-transfer, and fluid-flow aspects). Technical readers are encouraged to study their publications for further information.30−32 The SAGD process has performed successfully at the Underground Test Facility (UTF) in Alberta, Canada, such that a high and economical oil recovery rate has been attained. This thermal production technology appears to become more feasible technically and economically as great advancements in process control, drilling technologies, and operating conditions have been made recently, resulting in lower operating and well completion expenses.1,7,10,46,47

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EXPERIMENTAL WORK Figure 2 shows a simple schematic of the laboratory setup used in this study to operate the SAGD process. The setup included a physical model, a steam generator, a water/oil separator, a backpressure regulator, thermocouples, and pressure gauges. The porous model media had the following dimensions: height, 80− 120 cm; width, 30−50 cm; thickness, 4−8 cm. The fracture 1648



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PARTICLE SWARM OPTIMIZATION (PSO) A promising population-based metaheuristic known as particle swarm optimization (PSO) was proposed by Kennedy and Eberhart (1995).51 The PSO method is intended to simulate communal behavior (e.g., birds flocking) to attain defined objectives such as a promising position within a multidimensional space. The PSO algorithm has been successfully used in various science and engineering areas of application.51−54 As an evolutionary algorithm, PSO makes searches by employing a swarm containing particles that change over iterations. To obtain the optimal solution, each individual alters its search path based on two parameters, namely, its own best earlier practice, which is denoted as Pi, and the best practice of all other particles, denoted as Pg.51−54 A certain position in a D-dimensional space is assigned to each individual, and the status of each particle is described by its velocity and location. The velocity (Vti) and position (Xti) of particle i at iteration t are given by52−54

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

content (mol %) 0.66 0.23 10.35 2.35 1.95 11.50 15.10 57.86

N2 CO2 C1 C2 C3 C4−C5 C6−C11 C12+ property

value

Mw C12+ specific gravity (SG) at 60 °F resin content asphaltene content

485 g/gmol 1.0473 2.40 wt % 24.10 wt %

throughout the test trials. Other parameters such as pressure and temperature were measured at various parts of the experimental setup as well. When a sudden increase in the value of CSOR was experienced, the test was completed.

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n = 1, 2, ..., N

t Vit = [vit1 , vit2 , ..., viD ]

(2)

t Xit = [xit1 , xit2 , ..., xiD ]

(3)

It was found that no information about the previous velocities of the particles is available when the conventional PSO system is used. Shi and Eberhart proposed a new parameter, called inertia weight (ω), to overcome this drawback.52−54 Each individual modifies its velocity (vtij) to find the most favorable solution based on the relationship52−54

ARTIFICIAL NEURAL NETWORK (ANN) Artificial neural networks (ANNs) have been developed to mimic the task of the biological nodes of the brain. This technique has the strong ability to discover extremely multifaceted nonlinear correlations among parameters.48−50 The most common type of ANN used for modeling problems is the multilayer perceptron (MLP).48−50 The nonlinear input/output relationships in the MLP network are defined as follows:48−50 Let the parameters be of the form yn = fn (xn , ωn)

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vijt = ωvijt− 1 + C1r1(pijt − xijt) + C2r2(pgtj − xgtj) j = 1, 2, ..., D

(4)

where ω is the inertia weight and C1 and C2 represent the cognitive and social learning coefficients, respectively. r1 and r2 are two random parameters in the range of 0−1. In addition, Pti and Ptg are the local and global best solutions, respectively, and are expressed as52−54

(1)

in which yn and xn represent the output and input parameters, respectively. N is the number of processing nodes in the output layer. ωn is the vector of network weights.48−50 An MLP network generally includes three different layers, namely, input, hidden, and output layers, that can have various numbers of processing neurons. Each neuron in the input layer is associated with all of the neurons in the hidden and output layers through weighted bonds.48−50 The numbers of input and output neurons are usually determined according to the problem physics, available data, and targets defined by the researcher. However, the number of hidden neurons is considered to be flexible. Hence, it is obtained based on the network performance, convergence time, and closeness between the predicted and measured target parameters. The MLP network should be trained to determine the complex relationships between input and output parameters of the process (or problem) under study.48−50 In general, ANNs are categorized into two key groups, namely, supervised and unsupervised, depending on the training methods. The main criterion for obtaining the network parameters (e.g., optimum weights) during the training stage is minimization of the error function, which is the mean squared error (MSE). A trial-and-error technique is usually employed to determine the optimal momentum, learning cycle, number of hidden neurons, and learning rate. However, the optimum performance might not be attained because of the nature of data, data correctness, convergence criteria, and limitations of the computation process. The reader is encouraged to consult relevant publications for more information.48−50

t Pit = [pit1 , pit2 , ..., piD ] t t Pgt = [pg1 , pg2 , ..., pgtD ]

The linear function of the inertia weight is represented by ω = ωmax −

(ωmax − ωmin)t tmax

(5) (6) 52−54

(7)

where ωmax is the maximum magnitude of the inertia weight, ωmin is the initial magnitude of the inertia weight, t is the current iteration, and tmax is the total number of iterations. In general, a range of the particle velocity (e.g., [−vmax, vmax]) is specified in the PSO technique to keep the velocity at a reasonable level. The relationship between the previous best solution (xtij) and 52−54 the new best solution (xt+1 ij ) is given by xijt+ 1 = xijt + vijt

j = 1, 2, ..., D

(8)

The basic procedure of the PSO algorithm is provided as follows: (1) Initialize the system by creating initial individuals at random. (2) Determine the fitness of each individual in the swarm. (3) Update the velocity of each individual using eq 2. 1649



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course, the evaluation of other parameters is generally needed to properly design and operate a specific recovery technology under particular process and geological conditions. As an example, the effect of thermal conductivity on the design of a production strategy is not negligible; however, this property is not taken into account as a first-order screening parameter, as it does not have a wide range of values for different cases. Thus, it might not affect the decision to accept or reject a candidate EOR technique for a certain oil reservoir. It has been confirmed that just the first-order variables including fluid, geological, and petrophysical characteristics are needed to make an initial decision in the early stages of the screening process.10,71−74 To determine the important parameters influencing SAGD performance, we considered previous modeling and experimental approaches and also performed a comprehensive statistical investigation. These activities confirmed that the key parameters for ANN−PSO modeling to estimate RF and CSOR during the SAGD process consist of the steam injection rate, fracture permeability, matrix permeability, initial oil saturation, in situ viscosity, reservoir depth, effective porosity, steam quality, injection pressure, and reservoir thickness. It is clear that understanding of the physical concepts of the process can significantly assist in identifying these parameters. Development of the ANN−PSO Model. Before performing ANN−PSO modeling, important parameters such as oil viscosity, oil zone thickness (H), matrix permeability, and fracture permeability should be as the inputs for the connectionist modeling system, based on laboratory studies and the physics of the SAGD process in petroleum reservoirs (e.g., fractured media). Generally, an ANN involves four vital stages, including training, selection, testing, and validation. The training step is utilized to train the network, the selection process is used to choose proper parameters for the planned configuration, and the testing stage is used to guarantee that the system outputs after the selection step are generalized suitably. The final stage is the validation to examine the performance of the model. This stage also confirms that the developed model generates results that make sense technically while conducting parametric sensitivity analysis. In ANN modeling, the collected real data are split into two categories, for training and for testing. To make these parts, the data are chosen randomly. The number of training data points is also important, because, if a small population were picked, trustworthy outputs might not be achievable and the training process of the network would not be effective or reliable. Therefore, a proper choice can ensure high performance for both the training and testing stages. The main objective throughout the training process is to find the optimum magnitudes for the weights of the connections in the ANN structure layers. If the number of weights is larger than the number of presented data points, the error value in fitting the nontrained data is mostly reduced. The error might exhibit an increase as overtraining in the network occurs. In contrast, when the number of data points used is greater than the number of weights, the issue of overfitting matter is not at all important. The size of the data set employed in the training stage needs to be optimized to remove outliers for cases where decreasing the number of data points does not have a significant impact on the prediction accuracy. It is also important to note that each ANN structure typically consists of three layers, namely, input, hidden, and output layers. Optimizing the number of hidden neurons is normally carried out to reduce the time needed for the network overprediction procedure. PSO acts as an optimization

(4) Determine a new location for each particle based on eq 3. (5) If an assigned criterion is met, terminate the algorithm; otherwise, return to step 2. Further information about the theoretical concepts and implementation procedure of the PSO model is provided elsewhere.33,39,40,52−54

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METHODOLOGY Data Collection. A large number of real data points from SAGD experiments were collected to create an ANN−PSO model for predicting the SAGD recovery factor (RF) and cumulative steam-to-oil ratio (CSOR). The experimental results, numerical modeling outputs, and required data were obtained from studies conducted by several other researchers.19,21−25,55−70 The ranges of the input and output parameters used to build the ANN−PSO system are provided in Table 2. Table 2. Ranges of Parameters Used in the ANN Models to Estimate CSOR and RF property

minimum

maximum

1 0.1 11 0.3 100 165 50 25 6 1000

565 2000 38 1.0 1500 1485 400 90 2936 3000

3 35

11 72

Input matrix permeability (mDarcy) fracture permeability (Darcy) matrix porosity (%) steam quality steam rate (bbl/day) depth (m) reservoir thickness (m) initial oil saturation (%) oil viscosity (cP) injection pressure (psi) Output cumulative steam oil ratio recovery factor (%)

This table covers wide intervals of the important data, based on production history obtained from field trials, pilot-plant runs, and experimental tests of SAGD operations. If a data point were selected outside the ranges listed in Table 2, the methodology for conducting ANN−PSO modeling would be unchanged; however, a number of main parameters such as the magnitude of the learning factor and the number of hidden nodes should be varied such that a satisfactory match is obtained between the predicted and measured outputs. Selection of Input Variables. In a screening process, some of the available parameters are vital such that the planned assignment would be unfeasible if they were not considered in the study. These variables refer to first-order screening measures (or criteria).10,71,72 The critical (first) parameters strongly affect the choice for a particular objective, leading to change in the ranking of proper practical technologies throughout a general screening procedure. These factors also play an important role in the performance prediction of a certain technique. For instance, initial oil saturation exhibits a contribution of over 10% in all thermal EOR technologies. Therefore, it is placed in the category of first-order screening criteria in any assessment to select a production method for a particular viscous oil reservoir.10,71,72 It is important to note that first-order (critical) parameters are usually different for various processes or techniques. As an example, oil saturation, formation depth, net pay thickness, and oil viscosity are critical variables for most EOR technologies applied to heavy- and extra-heavy-oil reservoirs.10,71−74 Of 1650



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Figure 3. Simple design of a multilayer connectionist model (e.g., ANN) for the SAGD technique.

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RESULTS AND DISCUSSION In general, it is not possible to perform an extensive modeling simulation for most real cases, as the simulation study usually

algorithm, and MSE plays the role of a cost function when evaluating the network effectiveness through statistical criteria. The objective of the hybrid connectionist model is to minimize MSE while employing the optimization algorithm. As the first stage before applying the input and output data to the ANN−PSO model, we normalized the data, because the parameters (e.g., viscosity and permeability) differed in terms of type, units, and order of magnitude. To have all data between −1 and +1, the following relationship was used for normalization purposes Yi =

2(yi − ymin ) (ymax − ymin )

−1 (9)

where Yi represents the normalized magnitude of the variable yi. ymax and ymin represent the maximum and minimum values, respectively, of all used data. Assessment of the performance of the smart technique model was performed in this study using several statistical parameters, namely, mean squared error (MSE), maximum absolute percentage error (MAPE), minimum absolute percentage error (MIPE), and coefficient of determination (R2). The best ANN design for our SAGD case included 10 input neurons, 12 neurons in the first hidden layer, 20 neurons in the second hidden layer, and 2 output neurons. The back-propagation (BP) technique using the Levenberg−Marquardt method was employed for ANN training to predict the SAGD performance. The hidden and output layers in the current study had sigmoid and linear transfer functions, respectively. A schematic of the ANN configuration suggested for the current study is shown in Figure 3.

Figure 4. Recovery factor in terms of steam injection rate for different porous systems. Note: The fractures are far from the wells.

encounters high technical complexities and enormous computational times. Given these problems, a simple but accurate technique such as a smart model for estimating SAGD performance seems necessary. Certainly, if the method is applicable over wide ranges of important input variables, it can be employed as a proper tool to offer initial estimations at an early stage of field development without conducting complicated analytical modeling or/and numerical simulation. This section summarizes the results of the experimental work, statistical 1651



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Figure 5. Cumulative steam-to-oil ratio versus steam injection rate for different porous systems. Note: The fractures are far from the wells.

investigation, and developed hybrid smart technique applied to SAGD operation in petroleum reservoirs, with adequate discussion in a systematic way. Experimental Investigation. The experimental tests were performed with porous media with different permeabilities as various matrix and fracture parts were employed. To generalize the outcomes of the ANN−PSO system, a large number of laboratory and modeling works available in the open literature were utilized as well.19,21−25,55−70 The research studies addressed in this work also incorporate a range of fracture networks. The Supporting Information lists a portion of the real SAGD data employed. Because the actual data are gathered from various sources, wide ranges of oil and rock properties are covered in the current study. As a sample, production data for porous media (one homogeneous and two fractured systems) with a matrix permeability of 40 Darcy and a matrix porosity of 36% are given in this article. The fractured porous systems used contain two vertical fractures or two horizontal fractures placed above the injection well. It should be noted that the fractures are sufficiently far from the well pairs that the fracture network does not cause a rapid increase in the magnitude of SOR in the fractured porous media models. The fracture aperture is 2 mm for all fractures. Recovery factor (RF) is plotted as a function of pore volume of

Figure 6. Prediction of the CSOR in the SAGD process using ANN− BP, exhibited by scatter plots: (a) training phase, (b) testing phase.

injected steam in Figure 4 (above) for these three porous models undergoing the SAGD operation, based on cumulative oil production data. A comparison of the production data indicates that vertical fractures play an important role in extending quickly the area of depleted oil and temperature front because of the high

Table 3. Regression Models for the CSOR and RF Parameters CSORa

RFb

coefficient

numerical value

standard error

coefficient

numerical value

standard error

a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15

−17.731 −976.021 × 10−6 5.943 4300.365 6.013 × 10−6 −245.633 19.578 −39486.245 −3.761 2.045 × 10−3 −8.892 0.043 142865.813 19036.217 −1425.038 0.011

1.568 4.052 × 10−6 0.894 405.004 2.005 × 10−7 12.372 4.021 1708.554 0.314 7.115 × 10−5 2.201 4.675 × 10−3 10257.479 986.887 180.475 305.465 × 10−6

a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31

−79.763 0.694 142.851 0.044 0.337 295.345 −105.178 × 10−3 52.367 −25.425 × 10−5 −0.149 × 10−3 −0.048 −174.349 −0.127 9.825 × 10−4 −0.372 −0.011

12.648 0.083 23.546 0.003 0.027 27.940 648.976 × 10−4 3.842 6.119 × 10−6 0.027 × 10−3 0.005 48.672 0. 045 1.792 × 10−5 0.026 0.007

a 2

R = 0.969, F = 188.006; bR2 = 0.948, F = 84.852. 1652



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Figure 7. Estimation of the CSOR in the SAGD process using ANN− PSO through scatter plots: (a) training phase, (b) testing phase.

Figure 8. Performance assessment of ANN−BP in predicting RF using scatter plots: (a) training phase, (b) testing phase.

vertical conductivity of the fractures. Therefore, vertical fractures provide a higher probability for the steam to penetrate into the matrix medium, leading to greater oil recovery than for to the unfractured medium. In contrast, the presence of horizontal fractures limits the development of the steam chamber, resulting in a lower RF for the fractured medium (including just horizontal fractures) compared to the porous system with vertical fractures. This confirms that the presence of vertical fractures (and fracture networks) improves SAGD oil production, as it boosts the effective vertical permeability, enhances the cross-flow of steam and oil, and provides further area for steam diffusing into the bulk of the heavy oil within the matrix medium. Another conclusion is that the presence of fractures increases the effective permeability, leading to greater oil production as a result of the significant interaction between the matrix and fracture parts as well as the steam and oil contacts. As can be seen in Figure 4, the RF for the medium with vertical fractures, the medium with horizontal fractures, and the conventional porous medium (no fractures) are the highest, intermediate, and lowest ranks, respectively. Using the same logic, the porous model with vertical fractures and the conventional porous system produce the lowest and highest CSORs, respectively, as shown in Figure 5. In the case that the fractures are close enough to the production and injection wells, the reverse behavior is observed in terms of CSOR and RF as a function of pore volume (PV) of injected steam for these three models.

Empirical and Statistical Equations. Based on numerical simulation outputs obtained for a shallow Athabasca-type reservoir, a correlation was proposed by Shin to determine the cumulative steam-to-oil ratio (CSOR) in terms of reservoir and fluid properties as follows75 CSOR = 5.34(H 0.8K v 0.5So 2ϕ1.5)−0.52

(10)

where H is the oil zone thickness, Kv is the vertical permeability, So is the initial oil saturation, and ϕ represents the reservoir porosity. Shin also correlated the recovery factor (RF) with the important independent parameters in SAGD for two different conditions as follows75 RF = 0.09 ln(H1.8K v 0.8So1.2ϕ0.3) + 0.65 if (H1.8K v 0.8So1.2ϕ0.3) > 2.5

(11)

RF = 0.49ln(H1.8K v 0.8So1.2ϕ0.3) + 0.31 if (H1.8K v 0.8So1.2ϕ0.3) < 2.5

(12)

These equations are valid mostly for homogeneous heavy-oil reservoirs. They might provide good estimates for SAGD performance in heterogeneous media if the effective permeability and porosity are considered in the calculations of RF and CSOR. 1653



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Figure 9. Performance evaluation of ANN−PSO in estimating RF employing scatter plots: (a) training phase, (b) testing phase.

In addition, these correlations predict higher magnitudes for RF and CSOR compared to real data, as the simulation model considered the oil reservoirs to be ideal homogeneous media throughout the computation process. Based on experimental works and numerical studies, we developed the following empirical predictive functions in this study to predict CSOR and RF during SAGD operation in naturally fractured carbonate heavy-oil reservoirs. 1 1 1 CSOR = a0 + a1D + a 2 + a3 + a4K f + a5 ϕe Km μo 1 1 1 D + a6 + a 7 + a8xs + a 9Pinj + a10 + a11 So qs Soϕe H + a12

Figure 10. Accuracy of ANN−BP in estimating CSOR during the SAGD process with respect to closeness to line X = Y based on the value of R2: (a) training phase, (b) testing phase.

where D is the reservoir depth, qs is the steam injection rate, ϕe is the effective porosity, Km is the matrix permeability, Kf is the fracture permeability, xs is the steam quality, μo is the oil viscosity, and Pinj is the injection pressure. The magnitudes of the correlation coefficients and standard errors for eqs 13 and 14 are listed in Table 3. It should be noted here that F in the footnotes to Table 3 represents the degrees of freedom for the regression equations. For both RF and CSOR parameters, the magnitudes of F are greater than their critical values. Thus, these results confirm the significance and robustness of the predictive correlations developed in this study. It is not surprising that this study shows that the initial oil saturation and reservoir thickness are two important factors for screening heavy-oil reservoirs for the SAGD process. ANN−PSO Characteristics. This study included 170 data points to construct the ANN network. About 71% (120 data points) were assigned to the training stage, and the rest were considered for the testing phase. In general, there are some important parameters, including number of hidden neurons, number of particles, and learning coefficient in the ANN−PSO structure. The parameters were determined to be optimized to attain the outputs with reasonable accuracy for the specific case

K D 1 1 + a13 + a14 + a15 f HϕeK mμo Soqs ϕeqs ϕeK m Km (13)

RF = a16 + a17D + a18So + a19 H + a 20qs + a 21ϕe + a 22K m + a 23xs + a 24μo + a 25Pinj + a 26 + a 27Soϕe + a 28qsxs + a 29 + a31

Kf Km

SoϕeHK mqs μo xs

K mH μo

+ a30qsϕe

(14) 1654



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Figure 12. Performance of ANN−BP in predicting RF in terms of R2: (a) training phase, (b) testing phase.

2

Figure 11. Performance of ANN−PSO in predicting CSOR terms of R : (a) training phase, (b) testing phase.

it is not a rule in the PSO algorithm that C1 and C2 must have the same values, although this equality sometimes occurs in engineering and science problems. In fact, C1 and C2 are determined through an optimization approach as the best results for predicting the desired values of the target functions. In our particular case, the optimum values for C1 and C2 happened to be the same. It is clear that the training time for ANN systems is much longer than the time required to evaluate a correlation or/and empirical equation. Provided a 1844-bus system for the training stage, the CPU time for the conventional ANN (ANN−BP) is in the range of 25−35 min, whereas that for the ANN−PSO model is 45−75 min, depending on the number of iterations, first guesses, type of target parameter under study, and population size. Accuracy of the Model. To systematically examine the precision of the connectionist modeling introduced in this work, we employed statistical analysis and a graphical measure method. Statistical parameters including the mean squared error (MSE), minimum absolute percentage error (MIPE), maximum absolute percentage error (MAPE), and coefficient of determination (R2) were employed to check the closeness of the predicted SAGD performance parameters to the real outputs.76,77 The ANN−BP and ANN−PSO predictions for CSOR compared to the measured values are plotted in Figures 6 and 7, respectively, as

discussed in this study. The optimization procedure was performed a number of times to achieve the global optimum. The particle swarm optimization (PSO) technique was utilized in this study. The optimized magnitudes of the parameters for the ANN−PSO topology were obtained as follows: (1) Number of layers = 4, including 1 input layer, 2 hidden layers, and 1 output layer (2) Number of nodes (or neurons) = 10 in the input layer, 12 in the first hidden layer, 20 in the second hidden layer, and 2 in the output layer (3) Maximum number of iterations = 600 (4) C1 = C2 = 2.2 (5) Time interval = 0.015 s (6) Number of particles = 25 (7) Learning coefficient = 0.75 (8) Momentum correction factor = 0.001 (9) Initial inertia weight ωmin = 0.55 (10) Final inertia weight ωmax = 0.65 All of these parameters were obtained based on the network performance in terms of statistical criteria such as R2, MSE, MIPE, and MAPE. It is also important to note that C1 and C2 have different physical meanings (concepts) in the ANN−PSO system and that 1655



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Figure 14. Performance of the ANN−BP system as a function of number of epochs.

Figure 13. Precision of ANN−PSO in estimating RF based on the magnitude of R2: (a) training phase, (b) testing phase. Figure 15. Performance of the developed ANN−PSO model as a function of the number of epochs.

scatter plots for the training and testing stages. The same methodology was applied for the RF parameter in the SAGD process, as shown in Figures 8 and 9 for the ANN−BP and ANN−PSO systems, respectively. The scatter plots show the degree of agreement between the estimated values of the target parameters and the real data. It is clear from the figures that the existing smart models are suitable for predictive purposes within wide ranges of the input variables. Figures 10−13 display the relative errors between the predicted values and the corresponding real data for both stages in the connectionist modeling based on conventional ANN and ANN−PSO. The strength of the developed hybrid smart system (i.e., ANN−PSO) is noticeably

confirmed by the tight cloud of data with respect to the 45° line in examining the predictive tool for the training and testing steps (see Figures 11 and 13). We also compared the outputs obtained from the smart models (i.e., conventional ANN and ANN− PSO) to the data predicted using the correlations based on statistical parameters (i.e., MSE, MIPE, MAPE, and R2) as listed in Table 4. It is important to note that the error of the predictive tools in predicting RF and CSOR increases significantly for the heavy-oil reservoirs with fractures. As is clear from the table, the

Table 4. Performance of the Predictive Tools Applied to the SAGD Processa ANN−PSO RF

a

ANN−BP CSOR

RF

regression correlations CSOR

RF

CSOR

statistical parameter

Hom

Frac

Hom

Frac

Hom

Frac

Hom

Frac

Hom

Frac

Hom

Frac

R2 MSE MIPE (%) MAPE (%)

0.9896 0.0015 5.6341 10.3620

0.9785 0.0032 6.8879 11.0054

0.9904 0.0014 4.8765 9.3148

0.9847 0.0027 5.2463 10.2211

0.9783 0.0026 7.7884 11.4765

0.9602 0.0035 8.3241 12.3403

0.9821 0.0036 6.9085 11.1102

0.9745 0.0040 7.7653 12.4579

0.9535 0.0087 8.899 13.9674

0.9467 0.0096 9.7598 14.8465

0.9756 0.0079 7.4508 12.8794

0.9597 0.0158 8.9785 13.7978

Hom and Frac refer to homogenious and fractured porous media, respectively. 1656



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SAGD operation is implemented for heavy-oil underground formations. Figures 14 and 15 show the performance plots of the conventional ANN and ANN−PSO systems, respectively. The curves presented in the figures depict the connection between the training, testing, validation, and preeminent ANN models used for the estimation of SAGD performance in terms of MSE versus number of epochs. As indicated by the black circles in Figures 14 and 15, the best performance in terms of MSE is 0.0031 at about the fourth epoch for the validation phase of ANN−BP, whereas the hybrid ANN model achieves the greatest performance (MSE ≈ 0.0021) for the validation phase at the 11th epoch. Importance of Variables. According to Garson,76−78 a sensitivity analysis was carried out to find the relative effects (REs) of the input variables on the target functions for the ANN−PSO method when the values of the input and output linking weights are given. The corresponding equation is written as76−78

Figure 16. Relative impact of independent parameters on SAGD performance.

n ⎡⎛ ivj ⎞ ⎤ ∑ j =H 1 ⎢⎜ ∑nv i ⎟ωj ⎥ ⎣⎝ k =1 kj ⎠ ⎦ RE = ⎧ n ⎡⎛ ivj ⎞ ⎤⎫ n ∑i =v 1 ⎨∑ j =H 1 ⎢⎜ ∑nv i ⎟ωj ⎥⎬ ⎣⎝ k =1 kj ⎠ ⎦⎭ ⎩

(15)

in which nv and nH refer to the numbers of input neurons and hidden neurons, respectively. ωj presents the absolute magnitude of the linking weight between the output and hidden layers. The absolute magnitude of the input linking (connection) weights is defined by ivj. It is important to note that a greater value of RE for a certain input variable implies that the variable has a higher effect on the output parameter. The results of the sensitivity analysis are presented in Figure 16. As is clear from the figure, the initial oil saturation and reservoir thickness are the most important parameters among the inputs, contributing to the values of RF and CSOR. Increasing the initial oil saturation results in a higher RF and lower CSOR. Obviously, this outcome is in agreement with the results of other studies reported in the literature. The impacts of other variables on SAGD performance support the validity of the developed ANN−PSO and correlations introduced in this study as well. Validity of the Model. To check the validity of the introduced connectionist modeling, we conducted a simple parametric sensitivity analysis to study the effects of some important variables such as matrix permeability and reservoir thickness on the target variables (i.e., RF and CSOR). Then, we compared the obtained results with the measured output data. As depicted in Figures 17 and 18, the SAGD recovery factor improves with increasing permeability and reservoir thickness. On the other hand, increasing reservoir thickness and permeability decrease the magnitude of CSOR during SAGD operation. In addition, Figures 17 and 18 illustrate the changes in RF and CSOR with respect to permeability for both ANN−PSO and real data. As shown in the figures, the smart technique is able to successfully capture the relationship between the SAGD performance parameters (i.e., RF and CSOR) and independent variables such as permeability, porosity, and thickness. Case Study. The case study considered in this article is the Kuh-e-Mond Field, which is situated at Thrust Belt and Zagros Fold in the southern Iran. This field contains sedimentary layers with several seals and traps.79,80 Heavy-oil and gas reserves are generally found in this field. The first oil exploration in this

Figure 17. CSOR versus matrix permeability at various values of porosity (H = 22 m, So = 0.75, Kf = 5 Darcy, Pinj = 2500 kPa, xs = 0.95, qs = 300 m3/day, D = 540 m, μo= 23000 cP).

Figure 18. RF versus reservoir thickness at different matrix permeability (ϕe = 0.25, So = 0.75, Kf = 5 Darcy, Pinj = 2500 kPa, xs = 0.95, qs = 300 m3/day, D = 540 m, μo = 23000 cP).

developed ANN−PSO model has the lowest values of MSE, MIPE, and MAPE and the highest value of R2 for all types of petroleum reservoirs operating under SAGD. Again, this supports the robustness of the hybrid ANN (i.e., ANN−PSO) in estimating important parameters such as RF and CSOR when 1657



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Figure 19. Geographical site of the Kuh-e-Mond heavy-oil reservoir.

that the fractures mainly appear as vertical and subvertical in the reservoirs. The characteristics of the rocks and fluids of the field are listed in Table 5. Using the hybrid connectionist model (i.e., ANN−PSO), the ranges of CSOR and RF for the Jahrum heavy-oil reservoir were determined to be 5.0−6.3 and 50.2−56.3%, respectively. CSOR values of 5.9−7.4 and RF values in the range of 42.5−46.7% were estimated for the Sarvak heavy-oil reservoir for a steam injection rate and steam quality varying in the ranges of 700−900 bbl/day and 80−90%, respectively. This rapid screening study clearly demonstrates that it is theoretically possible to implement the SAGD thermal technique for production from both reservoirs in the Kuh-e-Mond Field. It is also concluded that, because it has more suitable formation characteristics, the Jahrum formation is a better candidate for SAGD than the Sarvak formation. Yet, oil production from this heavy-oil field is doubtful in terms of economic and operational aspects, implying that further theoretical and field studies are required before a decision on a production method for this field is made.

Table 5. Properties of Rocks and Fluids of the Kuh-e-Mond Field79,80 Reservoir property matrix permeability (mDarcy) fracture permeability (mDarcy) porosity (%) oil saturation (%) depth (m) thickness (m) net-to-gross ratio (%) net pay thickness (m) in situ oil viscosity (cP) in situ temperature (°C) wettability lithology

Sarvak

Jahrum

0.2−1.4

1.0

350−500

300−500

16 46 1100−1200 100 47 47 570 43.33 oil wet/mixed wet limestone

19 66 680−900 320 31 99 1160 21.11 oil wet dolomite and dolomitic limestone

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CONCLUSIONS Using the data produced from laboratory and pilot-plant test runs, as well as the results of numerical modeling studies, statistical correlations and a smart technique in the form of an ANN optimized with the PSO method were developed to predict the production performance of SAGD technology targeted for heavy-oil fractured reservoirs. The predictive tools consider RF and CSOR parameters as functions of various variables such as oil

district dates back to 1931. The types of formations in the Kuh-eMond field are fractured media, composed of carbonates.79,80 The Jahrum and Asmari Formations as Eocene carbonates and the Sarvak Formation as Cretaceous carbonates are the main parts of the field (see Figure 19). The fluid flow seems to be controlled by the natural fractures existing in the field, because the matrix permeability is very low.79,80 It should be noted here 1658



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AOSTRA = Alberta Oil Sands Technology Research Authority BP = backpropagation CMG = Computer Modeling Group CSOR = cumulative steam-to-oil ratio EOR = enhanced oil recovery MAPE = maximum absolute percentage error MIPE = minimum absolute percentage error MLP = multilayer perceptron MSE = mean squared error OIP = oil in place PSO = particle swarm optimization PV = pore volume RE = relative effect RF = recovery factor SAGD = steam-assisted gravity drainage SOR = steam-to-oil ratio UTF = underground test facility VAPEX = vapor extraction

saturation, oil viscosity, formation thickness, fracture permeability, and matrix permeability. On the basis of the results obtained from this study, the following important conclusions can be made: (1) The introduced hybrid ANN system (i.e., ANN−PSO) has substantial potential to estimate RF and CSOR for SAGD processes in heavy-oil reservoirs with sound precision. (2) The developed ANN−PSO model exhibits higher predictive performance than both the conventional neural network system (i.e., ANN−BP) and the regression equations obtained in this study. (3) Both experimental and connectionist modeling approaches demonstrated that the presence of fractures in highly fractured reservoirs leads to an increase in the magnitude of CSOR and, consequently, a reduction in RF throughout the implementation of SAGD. (4) The proposed ANN−PSO is not stuck in local minima when it is employed to predict SAGD production performance, because the algorithm includes both global and local searching properties. (5) The ANN−PSO system can be merged with heavy-oil recovery modeling software packages available for thermal EOR techniques to speed up the computational process, decrease the uncertainties, and improve the prediction and modeling capabilities. (6) The optimum ANN network contains 1 input layer (10 neurons), 2 hidden layers (12 neurons in the first layer and 20 neurons in the second layer), and 1 output layer (2 neurons), having acceptable statistical parameters including R2 and MSE values equal to 0.985 and 0.00074, respectively. (7) The suitable ANN configuration is generally determined using a trial-and-error technique. The development of a substitute method seems necessary to be joined to the PSO algorithm. (8) According to the relationship introduced by Garson, oil saturation and reservoir thickness were found to be the most important parameters affecting the process performance of SAGD processes.

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Variables

ai = coefficients for regression equations of RF and CSOR (i = 1, 2, ..., 31) C1 = cognitive learning coefficient in eq 4 C2 = social learning coefficient in eq 4 CSOR = cumulative steam-to-oil ratio D = reservoir depth (m or ft) F = degree of freedom H = oil zone thickness (m or ft) ivj = input linking (connection) weights K = intrinsic permeability (Darcy or m2) Kf = intrinsic permeability of the fracture (Darcy or m2) Km = permeability of the matrix in the model (Darcy or m2) Kv = vertical permeability in eq 10 (Darcy or m2) Mw = molecular weight (g/gmol) nH = number of hidden neurons nv = number of input neurons PV = pore volume (m3) P = pressure (atm or kPa) Pinj = injection pressure (atm or kPa) Pi = local best solution Pti = local best solution at iteration t Pg = global best solution Ptg = global best solution at iteration t ptij = local best solution at iteration t in the jth dimension ptgi = global best solution at iteration t in the jth dimension qs = steam injection rate (m3/day or bbl/day) R2 = coefficient of determination r1 = random parameter in eq 4 r2 = random parameter in eq 4 RE = relative effect RF = recovery factor S = fluid saturation So = oil saturation SG = specific gravity T = temperature (°C or °F) t = current iteration tmax = total number of iterations Vti = velocity of particle i at iteration t vtij = velocity of particle i at iteration t in the jth dimension Xti = position of particle i at iteration t xtij = position of particle i at iteration t in the jth dimension xn = input parameter in eq 1 xs = steam quality

ASSOCIATED CONTENT

S Supporting Information *

Part of the real data used in the current study. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Mitacs Elevate program and the Natural Sciences and Engineering Research Council of Canada (NSERC) for the financial support of this project.

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NOMENCLATURE

Acronyms

ANN = artificial neural network 1659



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(12) Rose, P. E. The Steam-Assisted Gravity Drainage of Oil Sand Bitumen; Ph.D. Dissertation, University of Utah, Salt Lake City, UT, Aug 1993. (13) Briggs, P. J.; Baron, R. P.; Fulleylove, R. J.; Wright, M. S. Development of heavy-oil reservoirs. J. Pet. Technol. 1988, 206−214. (14) International Energy Outlook 2011; U.S. Energy Information Administration: Washington, DC, Sep 2011. (15) Butler, R. M. Steam-assisted gravity drainage: Concept, development, performance and future. J. Can. Pet. Technol. 1994, 33 (2), 94-02-05. (16) Deng, X. Recovery performance and economics of steam/ propane hybrid process. Presented at the International Thermal Operations and Heavy Oil Symposium, Calgary, Alberta, Canada, Nov 1−3, 2005; SPE 97760. (17) McCormack, M. Mapping of the McMurray Formation for SAGD. J. Can. Pet. Technol. 2001, 39, 21−28. (18) Butler, R. M.; McNab, G. S.; Lo, H. Y. Theoretical studies on the gravity drainage of heavy oil during steam heating. Can. J. Chem. Eng. 1981, 59, 455−460. (19) Ito, Y.; Suzuki, S. Numerical simulation of the SAGD process in the Hangingstone oil sands reservoir. J. Can. Pet. Technol. 1999, 38 (9), 27−35. (20) Llaguno, P. E.; Moreno, F.; Garcia, R.; Mendez, Z. A reservoir screening methodology for SAGD applications. Presented at the Petroleum Society’s Canadian International Petroleum Conference, Calgary, Alberta, Canada, Jun 11−13, 2002; Paper 2002-124. (21) Shin, H.; Polikar, M. Review of reservoir parameters to optimize SAGD and fast-SAGD operating conditions. J. Can. Pet. Technol. 2007, 46 (1), 07-01-04. (22) Kamath, V. A.; Sinha, S.; Hatzignatiou, D. G. Simulation study of steam-assisted gravity drainage process in Ugnu Tar Sand Reservoir. Presented at the SPE Western Regional Meeting, Anchorage, AK, May 26−28, 1993; SPE 26075. (23) Kisman, K. E.; Yeung, K. C. Numerical study of SAGD process in the Burnt Lake Oil Sands Lease. Presented at the SPE International Heavy Oil Symposium, Calgary, Alberta, Canada, Jun 19−21, 1995; SPE 30276. (24) Ito, Y.; Suzuki, S.; Yamada, H. Effect of reservoir parameters on oil rates and steam oil ratios in SAGD projects. Presented at the Seventh UNITAR International Conference on Heavy Crude and Tar Sands, Beijing, China, Oct 27−30, 1988. (25) Sugianto, S.; Butler, R. M. The production of conventional heavy oil reservoirs with bottom water using steam-assisted gravity drainage. J. Can. Pet. Technol. 1990, 29 (2), 78−86. (26) Kashihara, K.; Takahashi, A.; Tsuji, T.; Torigoe, T.; Hosokoshi, K.; Endo, K. Geostatistical reservoir modeling focusing on the effect of mudstone clasts on permeability for the steam-assisted gravity drainage process in the Athabasca oil sands. In Heavy Oils: Reservoir Characterization and Production Monitoring; Chopra, S., Lines, L. R., Schmitt, D. R., Batzle, M. L., Eds.; Geophysical Developments Series; Society of Exploration Geophysicists (SEG): Tulsa, OK, 2010; Vol. 13, pp 203− 214. (27) Uwiera-Gartner, M. M. E.; Carlson, M. R.; Palmgren, C. T. S. Evaluation of the Clearwater Formation caprock for a proposed, low pressure, steam-assisted gravity-drainage pilot project in northeast Alberta. Presented at the SPE Annual Technical Conference and Exhibition, Denver, CO, Oct 30−Nov 2, 2011; SPE 147302. (28) Butler, R. M. Horizontal Wells for the Recovery of Oil, Gas and Bitumen; Petroleum Society Monograph 2; Prentice-Hall: Englewood Cliffs, NJ, 1994. (29) Butler, R. M. A new approach to the modeling of steam assisted gravity drainage. J. Can. Pet. Technol. 1985, 24 (3), 42−51. (30) Mohammadzadeh, O.; Chatzis, I. Pore-level investigation of heavy oil recovery using steam assisted gravity drainage (SAGD). Oil Gas Sci. Technol. 2010, 65 (6), 839−857. (31) Mohammadzadeh, O.; Rezaei, N.; Chatzis, I. Pore-level investigation of heavy oil and bitumen recovery using solvent-aided steam assisted gravity drainage (SA-SAGD) process. Energy Fuels 2010, 24 (12), 6327−6345.

Yi = normalized value of the variable yi ymax = maximum value of the variable y ymin = minimum value of the variable y yn = output parameter in eq 1 Greek Letters

μo = oil viscosity (cP) ϕ = porosity ϕe = effective porosity ω = inertia weight ωmax = maximum value of the inertia weight ωmin = minimum (or initial) value of the inertia weight ωn = network weight Subscripts

e = effective f = fracture inj = injection m = matrix max = maximum min = minimum o = oil s = steam v = vertical Metric Conversion Factors

°F = (°C × 1.8) + 32 1 bbl of oil = 0.159 m3 1 psi = 6.8947 kPa 1 cP = 0.001 kg/m·s

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REFERENCES

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