DSpace at VNU: Application of D-optimal Design for Modeling and Optimization of Operation Conditions in SAGD Process

This article was downloaded by: [Universidad Autonoma de Barcelona] On: 28 October 2014, At: 02:22 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Energy Sources, Part A: Recovery, Utilization, and Environmental Effects Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ueso20 Application of D-optimal Design for Modeling and Optimization of Operation Conditions in SAGD Process ac a b a H X Nguyen , Wisup Bae , W S Ryoo , M J Nam & T N Tu a a Department of Energy and Mineral Resources Engineering, Sejong University, Seoul, Korea b Materials Engineering and Sciences Division, Hongik University, Seoul, Korea c Faculty of Geology and Petroleum Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam Published online: 24 Jul 2014 To cite this article: H X Nguyen, Wisup Bae, W S Ryoo, M J Nam & T N Tu (2014) Application of D-optimal Design for Modeling and Optimization of Operation Conditions in SAGD Process, Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 36:19, 2142-2153, DOI: 10.1080/15567036.2011.557706 To link to this article: http://dx.doi.org/10.1080/15567036.2011.557706 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Energy Sources, Part A, 36:2142–2153, 2014 Copyright © Taylor & Francis Group, LLC ISSN: 1556-7036 print/1556-7230 online DOI: 10.1080/15567036.2011.557706 Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 Application of D-optimal Design for Modeling and Optimization of Operation Conditions in SAGD Process H X Nguyen,1;3 Wisup Bae,1 W S Ryoo,2 M J Nam,1 and T N Tu1 Department of Energy and Mineral Resources Engineering, Sejong University, Seoul, Korea Materials Engineering and Sciences Division, Hongik University, Seoul, Korea Faculty of Geology and Petroleum Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam The aim of this research is to apply experimental design methodology for the optimization and sensitivity analysis of operating parameters on the steam-assisted gravity drainage process These experiments, consisting of 26 cases, are determined by the D-optimal design A response surface method was used to maximize net present value, which was obtained at 262.42 $mm for an optimal operation condition The predicted values matched the experimental values reasonably well with R2 of 0.987 and Q of 0.902 for the NPV response Keywords: D-optimal design, net present value, optimization, response surface methodology, steamassisted gravity drainage INTRODUCTION A huge quantity of heavy oil and bitumen resources has been discovered worldwide According to Chen et al.’s research (2008), the proved heavy oil reserves are estimated at more than 1.8 trillion bbl in Venezuela, 1.74 trillion bbl in Alberta, Canada, and 20 to 25 billion bbl on the North Slope of Alaska However, extremely high viscosity of bitumen at normal temperatures is one of the biggest challenges for the recovery process The steam-assisted gravity drainage (SAGD) process is an effective method for heavy oil and bitumen production utilizing two parallel horizontal wells, one above the other As steam is continuously injected in the upper well, a steam chamber forms in the reservoir and grows upward to its surroundings, displacing heated oil following a gravitymechanism drain into the producer (Butler, 2001) However, the economic aspect of the SAGD process is risky because of the high capital cost for building ground facilities and uncertainties of oil and gas prices In order to predict production performance and profitability, optimal operation conditions should be determined by reservoir simulations The target is to maximize net present value (NPV) of the SAGD process, which is significantly affected by operating condition of injector/producer spacing (IPS), injection pressure (IP), maximum steam injection rate (MSIR), and spacing between two well pairs (WPS) Address correspondence to Prof Wisup Bae, Sejong University, 98 Gunja-dong, Gwangjin-ku, Seoul 143-747, Korea E-mail: wsbae@sejong.ac.kr Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ueso 2142 Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 APPLICATION OF D-OPTIMAL DESIGN 2143 Polikar et al (2000), Gong et al (2002), and Shin and Polikar (2007) have conducted optimization by classical methods based on their numerical simulations and experiments However, there is a lack of confidence level in the optimized conditions because they did not determine the significance level of operational parameters and ignored interactions’ effects between considered parameters, which may lead to low efficiency issues in the SAGD operation These limitations of the classical method can be avoided by applying D-optimal design and response surface methodology (RSM) that involves statistical design of experiments in which all factors are varied together over a set of experimental runs In addition, economic models in previous studies were not comprehensive enough with limited consideration on only three factors: low steam cost, low bitumen price, and discount rate That approach reduced the accuracy of economic evaluation and operation efficiency in practice In this study, D-optimal design and response surface methodology were applied for optimizing operation condition and mitigating economic risk A two-stage approach was employed for efficient local optimization First, an initial sample of design was obtained by using D-optimal design The responses for design points were estimated by amount of oil recovery and NPV Second, RSM was used to search for optimal designs The uncertainty in NPV was evaluated, and the optimization operating conditions of maximizing NPV was identified by building a surface response map RESEARCH METHODOLOGY 2.1 Basic Theory of Response Surface Methodology Response surface methodology is a collection of statistical and mathematical methods that are useful for designing experiments, building models, evaluating the effect of factors, and searching for optimum conditions for desirable responses (Box and Wilson, 1951) The RSM technique can improve product yields and provide closer confirmation of the output response toward the nominal and target requirements In recent years, RSM played an important role in oil fields, especially applications into enhanced oil recovery In most RSM problems, the objective function of the response and independent variables is unknown Thus, the first step is to find a suitable approximation for the true functional relationship between the response Y / and the set of independent variables Xi / If the response is well modeled by a linear function of the independent variables, then the approximation function is the first-order model: Y D ˇ0 C ˇ1 X1 C ˇ2 X2 C ˇ3 X3 C : : : C ˇk Xk C "; (1) where X1 ; X2 ; : : : ; Xk are the independent variables, ˇ0 is the constant coefficient, ˇk is the linear effect of the kth factor coefficients, and " is the error observed in the response Y If there is curvature in the system, then a polynomial of a higher degree must be used, such as the secondorder model (Myers et al., 2008): Y D ˇ0 C k X i D1 ˇi Xi C k X ˇi i Xi2 C i D1 X ˇij Xi Xj C "; (2) i 0) Amount of injected steam and water-handling costs significantly impact the NPV Based on the D-optimal design, the objective function is determined by a RSM that showed the correlation between the responses of NPV and a set of four operating variables of SAGD process, namely, spacing between injector and producer (IPS, X1 ), injection pressure (IP, X2 ), maximum steam injection rate (MSIR, X3 ), and SAGD well pattern spacing (WPS, X4 ) The number of tests required for the four independent variables are 26 cases in Table 2, in which both coded and actual levels of the variables in the design matrix were calculated With the effects of the interactions between two-factor and main factors included, Eq (2) can be rewritten as: Y D ˇ0 C ˇ1 X1 C ˇ2 X2 C ˇ3 X3 C ˇ4 X4 C ˇ11 X12 C ˇ22 X12 C ˇ33 X32 C ˇ44 X42 C ˇ12 X1 X2 (6) C ˇ13 X1 X3 C ˇ14 X1 X4 C ˇ23 X2 X3 C ˇ24 X2 X4 C ˇ34 X3 X4 : The coefficients of the main effects ˇi and two-factor interactions ˇij / were estimated from the experimental data obtained by computer simulation programming utilizing least squares method of @R12.2.1 software 2146 H X NGUYEN ET AL TABLE Independent Variables and the Result of D-optimal Design for SAGD Operation Coded Level of Variables Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 Run 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 X1 1 1 1 1 1 1 1 1 1 1 1/3 1/3 1/3 1/3 0 X2 1 1 1 1 1 1 1/3 1/3 1 1/3 1/3 1/3 1 1 0 X3 X4 1 1 1 1 1/3 1 1/3 1 1/3 1/3 1 1 1 0 1 1 1 1/3 1/3 1/3 1 1/3 1 1 1 1 0 Response (Simulation Observed) Actual Level of Variable IPS, m IP, kPa MSIR, m3 /d WPS, m Oil Cum., bbl NPV, $mm 17 5 17 17 17 5 5 5 17 17 17 17 17 17 13 9 13 11 11 1,500 1,500 4,500 1,500 1,500 4,500 1,500 1,500 1,500 4,500 4,500 4,500 3,500 2,500 4,500 4,500 4,500 3,500 2,500 3,500 1,500 4,500 4,500 4,500 3,000 3,000 360 360 840 360 360 360 840 840 680 360 360 520 840 840 840 520 680 360 840 840 840 360 840 840 600 600 40 40 40 160 160 160 160 120 40 80 120 160 40 160 80 40 40 40 40 160 40 40 160 160 100 100 6,968,044 5,974,498 7,374,708 6,954,205 4,970,181 7,012,793 5,637,592 7,279,099 7,087,982 7,232,437 7,236,378 7,300,388 7,376,898 7,308,108 7,386,310 7,256,499 7,303,452 7,110,347 7,270,746 7,321,858 6,236,887 7,129,522 7,372,203 7,346,328 7,358,968 7,358,968 196.88 133.46 252.37 194.65 102.38 192.53 120.04 239.29 225.70 203.04 203.17 238.83 255.39 256.13 244.39 227.09 232.64 202.91 185.16 234.45 141.56 209.55 251.52 242.98 245.44 245.44 3.3 Model Adequacy To prove the accuracy of a primary model, statistical analysis techniques were checked by the experimental error, the suitability of the model, and the statistical significance of the terms in the model The quality of the model is statistically measured by examining R2 , Radj , and Q2 The coefficient of multiple determination R is considered as the percentage of variability observed on the response and can be explained by the suitability of the regression model An adjusted coefficient Radj is used to compare the qualities of different models Any model with values of 2 both R and Radj close to indicates an excellent quality in fitting the observed data However, it 2 noted that both R and Radj does not give any information about its power of prediction between available data points In some cases, models that properly fit the experimental data may not have a good predictability In order to define the power of prediction of a model, using Q2 is measured based on the prediction sum of squares (PRESS) PRESS is a sum of squared differences between observed Y and predicted value Ypred The minimization of PRESS leads to an improvement on the power of prediction of the model Once a model is constructed as a result of the above consequences, it can be used to predict reservoir performance and to optimize controllable variables (Vanegas and Cunha, 2008) APPLICATION OF D-OPTIMAL DESIGN 2147 TABLE Regression Coefficients of the Predicted Quadratic Polynomial Model Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 NPV Estimate Constant IPS IP MSIR WPS IPS*IPS IP*IP MSIR*MSIR WPS*WPS IPS*IP IPS*MSIR IPS*WPS IP*MSIR IP*WPS MSIR*WPS Standard Error P Conf int(˙) 244.79 5.190 4.77E-14 24.63 1.752 2.26E-08 29.38 1.782 4.20E-09 15.14 1.786 3.74E-06 0.49 1.732 0.783025 3.40 4.305 0.445785 27.68 3.953 2.26E-05 16.85 4.278 0.002318 10.77 4.329 0.030135 19.83 2.002 8.13E-07 5.08 1.921 0.022777 2.32 1.918 0.251942 8.50 1.961 0.001186 1.38 1.985 0.500172 6.20 1.945 0.008681 Confidence level D 95% 11.424 3.857 3.923 3.931 3.811 9.474 8.701 9.415 9.529 4.407 4.229 4.222 4.315 4.369 4.282 RESULT AND DISCUSSIONS There are 26 scenarios for optimizing the four parameters in Table The result showed that cumulative oil of case 15 was the highest, but its NPV was lower than the others were Meanwhile, the highest NPV of 256.13 $mm was recorded in case 14 under the operation conditions of IPS m, IP 2,500 kPa, MSIR 840 m3 /d, and WPS 160 m Parameters of a quadratic polynomial model computed from experimental runs, the main effects ˇi /, and two-factor interactions ˇij / for four independent variables are presented in Table Consequently, the polynomial model describing the correlation between overall response and the variables can be rewritten as: NPV D 244:79 24:63X1 C 29:38X2 C 15:14X3 16:85X32 10:77X42 C 19:83X1 X2 0:49X4 C 3:4X12 5:08X1 X3 27:68X22 2:32X1 X4 C 8:5X2 X3 (7) C 1:38X2 X4 C 6:2X3 X4 : The analysis of variance, goodness-of-fit, and the adequacy of the model were summarized in Table The determination coefficient R2 is 0.987 The adjusted determination coefficient RAdj D 0:97/ confirmed that the model has high quality in fitting experimental data A very low value of coefficient of residual standard deviation (RSD D 7.55) clearly indicated a high degree of precision and reliability of the experimental values and in relation to the power of prediction, Q2 D 0:902 4.1 Quantitative Effects of Operating Parameters on the NPV Student’s t-test performs quantitative effects of the main factors The regression coefficient of Eq (7) showed standard errors and p-values (Table 3) The p-values are used to check the significance coefficient, which in turn may indicate the pattern of interactions between variables It can be seen that the linear coefficients of IPS, IP, and MSIR; the quadratic term coefficients of IP2 , MSIR2 , and 2148 H X NGUYEN ET AL Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 TABLE ANOVA Analysis NPV DF Total Constant Total corrected Regression Residual N D 26 DF D 11 26 25 14 11 SS 1,201,250 1,153,740 47,509.3 46,882.4 626.841 RAdj D 0.97 R D 0.987 Q2 D 0.902 MS 46,201.9 1,153,740 1,900.37 3,348.74 56.9856 F p SD 58.7648 43.5932 57.8683 7.54888 Cond no D 7.77 RSD D 7.55 WPS2 ; and the cross product coefficients of IPS.IP, IPS.MSIR, IP.MSIR, and MSIR.WPS were significant, with very small p-values (p < 0.05) Coefficients of other terms were not significant p > 0:05/ It is noteworthy that a positive sign indicates a synergistic effect, while a negative sign represents an antagonistic effect of a factor on the selected response 4.2 Main and Interaction Effect Plots The main effect plot is a useful tool for analyzing data of the relative significance of each factor in a design model It helps to identify important factors that significantly affect overall outcome, especially when the factors are at two or more levels Figure illustrated the effect level of each factor in the polynomial model of Eq (7) This Pareto graph is divided into two regions The region below zero, negative coefficients (IP.IP, IPS, MSIR.MSIR, WPS.WPS, IPS.MSIR, IPS.WPS, and WPS), indicated that an increase in the single and combination factors decreased on the NPV The region above zero, positive coefficients (IP, IPS.IP, MSIR, IP.MSIR, MSIR.WPS, IPS.IPS, and IP.WPS), indicated increase of NPV with an increase of the factors Single factors of injection pressure, injector producer spacing, and maximum steam injection rate have significantly affected the NPV FIGURE The interaction effect of operating parameters on the NPV Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 APPLICATION OF D-OPTIMAL DESIGN FIGURE 2149 Effects of main factors on the NPV Figure indicated that injector-producer spacing of m was the best design to maximum NPV with lowest cumulative steam oil ratio (CSOR) at the same time NPV reduces when IPS is over m because the preheating period extends a long time, which delays SAGD operation Similarly, an optimal condition for injection pressure, steam injection rate, and well pattern spacing is designed in the vicinity of 3,500 kPa, 750 m3 /d, and 120 m, respectively 4.3 Optimization of Operating Condition for SAGD Process The full model of objective function is given in Eq (7) The graphical representations of NPV contour and response surface plots were shown in Figures and 4, respectively The value of predicted maximum on the surface is confined in the smallest ellipse in the contour diagram Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 2150 H X NGUYEN ET AL FIGURE 2-D contour plots showing the effects of operation parameters on the NPV Elliptical contours are obtained, in general, when there is a perfect interaction between the independent variables The values of independent variables in the smallest contour and the corresponding maximum are determined to be the optimal operating conditions and the response of the dependent variable Optimal operating conditions showed at red smallest region where maximum NPV achieved over 281 $mm (Figures 3d and 4d) The favorable design for WPS, MSIR, IPS, and IP correspond to 120 m, 774 m3 /d, m, and 3,427 kPa, respectively (Figure 5) After a preheating period of 65 days, the SAGD process started to operate The growth of the steam chamber reaches up to the top reservoir in 230 days At the same time, heat is lost at overburden Until 750 days, steam chambers connected to be a unique amount of oil recovery, which were also obtained at maximum value Production performance with time is shown in Figure 4.4 Test of Predictive Model The validity for predicting optimum response is rechecked under an operating condition with IPS m, IP 3,427 kPa, MSIR 774 m3 /d, and WPS 120 m This design for the SAGD process was used to validate the model by comparing the predicted values of the responses to the results of experimental simulation Amount of oil recovery is about 7,414,537 bbl (Figure 7) in the simulated Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 APPLICATION OF D-OPTIMAL DESIGN FIGURE 3-D contour plots showing the effects of operation parameters on the NPV FIGURE Steam chamber performance in optimal operating condition 2151 Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 2152 H X NGUYEN ET AL FIGURE FIGURE Oil recovery in SAGD process Production performance in optimal condition operation of SAGD process Peak oil was obtained at 2,257,023 bbl in the second year, and then dramatically reduced until operation ended NPV is achieved at 262.42 $mm as the highest NPV among experimental cases in Table The result demonstrated the validity of the RSM model, which is reasonably adequate to predict the performance of SAGD operation (Table 5) TABLE Observed and Predicted Values under Optimal Conditions IPS, m IP, kPa MSIR, m3 /d WPS, m NPV, $mm (Predicted) NPV, $mm (Observed) Cum Oil, bbl Difference, % 3,427 774 120 281.3 262.42 7,414,537 APPLICATION OF D-OPTIMAL DESIGN 2153 Downloaded by [Universidad Autonoma de Barcelona] at 02:22 28 October 2014 CONCLUSIONS The application of a mathematical model and the optimization on the basis of statistical design of experiments is proven to be a useful tool to predict and analyze the interaction effects between operating factors Response surface methodology and D-optimal design were successfully applied to determine the optimal conditions in the SAGD process A maximum NPV of 262.42 $mm was obtained for an optimal operation design: injector-producer spacing of m, injection pressure of 3,427 kPa, maximum steam injection rate of 774 m3 /d, and spacing between two well pairs of 120 m The predicted values matched the experimental values reasonably well with R2 of 0.987 and Q2 of 0.97 for NPV response The D-optimal design with RSM not only improves the economic feasibility of bitumen recovery process but provides an economical way of obtaining the maximum profit in a short period of time with the fewest number of experiments ACKNOWLEDGMENT The authors wish to thank K K Schlumberger for the encouragement in writing this article FUNDING This work was supported by the Energy Resources R&D program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No 20102020300090) REFERENCES Box, G E P., and Wilson, K B 1951 On the experimental attainment of optimum conditions J Roy Stat Soc., Ser A 13:1–45 Butler, R M 2001 Some recent development in SAGD J Can Pet Technol 40:18–22 Canadian National Energy Board 2006 Canada’s Oil sands: Opportunities and Challenges to 2015 Calgary: Canadian National Energy Board Chen, Q., Gerritsen, M G., and Kovscek, A R 2008 Effects of reservoir heterogeneities on the steam assisted gravity drainage process SPE Reserv Eval & Eng 11:921–932 Gong, J., Polikar, M., and Chalaturnyk, R J 2002 Fast SAGD and geomechanical mechanism Paper CIPC 2002-163 Canadian International Petroleum Conference, Calgary, Canada, June 11–13 Myers, R H., Montgomery, D C., and Anderson-Cook, C 2008 Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Ed New York: John Wiley and Sons, pp 13–135 Polikar, M., Cyr, T J., and Coates, R M 2000 Fast SAGD: Half the wells and 30% less steam Paper SPE 65509 International Conference on Horizontal Well Technology, Calgary, Canada, November 6–8 Shin, H., and Polikar, M 2007 Review of reservoir parameters to optimize SAGD and fast-SAGD operating conditions J Can Pet Technol 46:35–41 Vanegas, J W., and Cunha, L B 2008 Prediction of SAGD performance using response surface correlations developed by experimental design techniques J Can Pet Technol 47:58–64  ... collection of statistical and mathematical methods that are useful for designing experiments, building models, evaluating the effect of factors, and searching for optimum conditions for desirable... applied for optimizing operation condition and mitigating economic risk A two-stage approach was employed for efficient local optimization First, an initial sample of design was obtained by using D-optimal. .. to determine the optimal conditions in the SAGD process A maximum NPV of 262.42 $mm was obtained for an optimal operation design: injector-producer spacing of m, injection pressure of 3,427 kPa,