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

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Energy Sources, Part A: Recovery, Utilization, and Environmental Effects

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Application of D-optimal Design for Modeling and Optimization of Operation Conditions in SAGD Process

H X Nguyenac, Wisup Baea, W S Ryoob, M J Nama & T N Tua 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

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Energy Sources, Part A, 36:2142–2153, 2014

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ISSN: 1556-7036 print/1556-7230 online

DOI: 10.1080/15567036.2011.557706

Application of D-optimal Design for Modeling and Optimization of Operation Conditions in SAGD Process

H X Nguyen,1;3Wisup Bae,1 W S Ryoo,2 M J Nam,1 and T N Tu1

1Department of Energy and Mineral Resources Engineering, Sejong University,

Seoul, Korea

2Materials Engineering and Sciences Division, Hongik University, Seoul, Korea

3Faculty 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 Q2of 0.902 for the NPV response

Keywords: D-optimal design, net present value, optimization, response surface methodology, steam-assisted gravity drainage

1 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 gravity-mechanism 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

Polikar et al (2000), Gong et al (2002), and Shin and Polikar (2007) have conducted opti-mization 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

2 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 ˇ0C ˇ1X1C ˇ2X2C ˇ3X3C : : : C ˇkXkC "; (1) where X1; X2; : : : ; Xkare the independent variables, ˇ0is the constant coefficient, ˇkis 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 second-order model (Myers et al., 2008):

Y D ˇ0C

k

X

ˇiXi C

k

X

ˇi iX2

i <j

ˇijXiXj C "; (2)

where ˇij represents the quadratic effect of the i th factor, and ˇij represents the cross product effect or interaction effect between the i th and j th factors The ˇ coefficients, which should be determined in the second-order model, are obtained by the least squares method In general, Eq (2) can be written in matrix form:

2144 H X NGUYEN ET AL.

where Y is defined to be a matrix of measured values, and X is a matrix of independent variables The matrices ˇ and " represent coefficients and errors, respectively Equation (3) can be rearranged for coefficients matrix ˇ by the matrix approach:

ˇ D X0:X/ 1X0Y; (4) where X0 is the transpose of the matrix X, and X0:X/ 1 is the inverse of the matrix X0:X The coefficients, i.e., the main effect ˇi/and two-factor interactions ˇij/, can be estimated from experimental results by computer simulation programming with the least squares method using

@R12.2 software

Using RSM is to predict the responses and to determine factors that optimized the second-order function An important assumption of the independent variables is continuous and controllable by experiments with negligible errors The task then is to find a suitable approximation for the true functional relationship between independent variables and the response surface

2.2 D-optimal Design

There are several design optimality criteria needed to do a computer-generated design, such as D-optimality, A-optimality, and G-optimality (Myers, 2008) Among them, the D-optimality is the most popular one and it is applied in this study In general, modeling accuracy, goodness-of-fit, can be statistically measured by a variance-covariance matrix V b/:

V b/ D 2.X0X/ 1; (5) where  is the standard deviation, an accurate response surface model is obtained when minimizing X0X/ 1 Statistically, minimizing X0X/ 1 is equivalent to maximize the determinant of X0X This criterion is to generate a design matrix with a maximized jX0Xj from a set of candidate samples, which can be defined by the D-optimality The initial “D” stands for ‘determinant’ By using D-optimal design, the generalized variance of a predefined model is minimized, which means the ‘optimality’ of a specific D-optimal design is model dependent Unlike conventional designs, D-optimal designs are straight optimization and their matrices are generally not orthogonal with the effect estimates correlated by variables

In D-optimal design, two coded levels (˙1) showed the lower and the upper bounds of input parameter The bounds should be chosen in an acceptable value range Then, an analyst must predefine the model type with consideration of linearity and interaction effects, which affects the sample number It is noted that for deterministic experiments, replicate samples are unnecessary because of no measurement errors in numerical experiments The level combinations of input parameters are generated from a larger candidate set as inputs for finite element analysis to obtain responses Finally, a response surface model results by estimating the coefficients ˇij in Eq (2) using the least square algorithm

3 THE D-OPTIMAL DESIGN FOR SAGD OPERATING CONDITION

3.1 Reservoir Model

The physical model corresponds to grid cell 201 m  900 m  30 m with no aquifer belonging

to McMurray formation, Athabasca oil sands The SAGD model is designed with two horizontal wells, one injector located above the other oil producer Steam is injected continuously into the heated bitumen reservoir causing oil flow into the producer Reservoir fluids were modeled as oleic, gaseous, and aqueous phases corresponding to three components of hydrocarbon, gas, and

TABLE 1 Reservoir Properties in McMurray Formation, Athabasca Parameters McMurray Formation Reservoir size 201 m  900 m  30 m Reservoir pressure, kPa 1,500

Depth to top of reservoir, m 200 Reservoir thickness, m 30 Reservoir width, m 201 Vertical permeability (Kv), Darcy 5 Permeability ratio (Kh/Kv) 2.5

Oil saturation 0.8 Oil viscosity at reservoir temperature 2,000,000 Reservoir temperature, ı C 12 Steam quality 0.95

water, respectively Grid, rocks, and fluid properties used in the simulation model are listed in Table 1

3.2 Economic Model

Economic evaluation is designed on the previous discussion in the Canadian National Energy Board reports (2006; versions 2006, 2008) Net present value was selected as the response variable

to measure the SAGD performance, and therefore the dependent variable in the proposed surface response correlation Net present value is computed based on the time cash flow with 10% yearly discount rate during 10 years of production phase The input parameters include oil rate, steam injection rate, and water produced rate The average prices of bitumen and gas are $70/bbl and

$4/mcf, respectively Drilling and completion costs of a SAGD well pair are $5.0 mm considered

as capital investment Total operating costs comprised of the electric cost of $1.0 per barrel of produced oil, water handling cost of $2/bbl, non-gas cost of $6/bbl, emission cost of $1/bbl, and field insurance of $0.5/bbl The feasible economics must be at least at 36,000 bbl/year for a SAGD well pair (NPV > 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 ˇ0C ˇ1X1C ˇ2X2C ˇ3X3C ˇ4X4C ˇ11X12C ˇ22X12C ˇ33X32C ˇ44X42C ˇ12X1X2

C ˇ13X1X3C ˇ14X1X4C ˇ23X2X3C ˇ24X2X4C ˇ34X3X4:

(6)

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 2 Independent Variables and the Result of D-optimal Design for SAGD Operation

Coded Level of Variables Actual Level of Variable

Response (Simulation Observed) Run X 1 X 2 X 3 X 4 IPS, m IP, kPa MSIR, m 3 /d WPS, m Oil Cum., bbl NPV, $mm

1 1 1 1 1 5 1,500 360 40 6,968,044 196.88

2 1 1 1 1 17 1,500 360 40 5,974,498 133.46

3 1 1 1 1 5 4,500 840 40 7,374,708 252.37

4 1 1 1 1 5 1,500 360 160 6,954,205 194.65

5 1 1 1 1 17 1,500 360 160 4,970,181 102.38

6 1 1 1 1 17 4,500 360 160 7,012,793 192.53

7 1 1 1 1 17 1,500 840 160 5,637,592 120.04

8 1 1 1 1/3 5 1,500 840 120 7,279,099 239.29

9 1 1 1/3 1 5 1,500 680 40 7,087,982 225.70

10 1 1 1 1/3 5 4,500 360 80 7,232,437 203.04

11 1 1 1 1/3 5 4,500 360 120 7,236,378 203.17

12 1 1 1/3 1 5 4,500 520 160 7,300,388 238.83

13 1 1/3 1 1 5 3,500 840 40 7,376,898 255.39

14 1 1/3 1 1 5 2,500 840 160 7,308,108 256.13

15 1 1 1 1/3 17 4,500 840 80 7,386,310 244.39

16 1 1 1/3 1 17 4,500 520 40 7,256,499 227.09

17 1 1 1/3 1 17 4,500 680 40 7,303,452 232.64

18 1 1/3 1 1 17 3,500 360 40 7,110,347 202.91

19 1 1/3 1 1 17 2,500 840 40 7,270,746 185.16

20 1 1/3 1 1 17 3,500 840 160 7,321,858 234.45

21 1/3 1 1 1 13 1,500 840 40 6,236,887 141.56

22 1/3 1 1 1 9 4,500 360 40 7,129,522 209.55

23 1/3 1 1 1 9 4,500 840 160 7,372,203 251.52

24 1/3 1 1 1 13 4,500 840 160 7,346,328 242.98

25 0 0 0 0 11 3,000 600 100 7,358,968 245.44

26 0 0 0 0 11 3,000 600 100 7,358,968 245.44

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, R2

adj, and Q2 The coefficient of multiple determination R2 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 R2

adj is used to compare the qualities of different models Any model with values of both R2and R2

adjclose to 1 indicates an excellent quality in fitting the observed data However, it noted that both R2 and R2

adjdoes 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 Q2is 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)

TABLE 3 Regression Coefficients of the Predicted Quadratic Polynomial Model

NPV Estimate

Standard Error P Conf int(˙) Constant 244.79 5.190 4.77E-14 11.424 IPS 24.63 1.752 2.26E-08 3.857

IP 29.38 1.782 4.20E-09 3.923 MSIR 15.14 1.786 3.74E-06 3.931 WPS 0.49 1.732 0.783025 3.811 IPS*IPS 3.40 4.305 0.445785 9.474 IP*IP 27.68 3.953 2.26E-05 8.701 MSIR*MSIR 16.85 4.278 0.002318 9.415 WPS*WPS 10.77 4.329 0.030135 9.529 IPS*IP 19.83 2.002 8.13E-07 4.407 IPS*MSIR 5.08 1.921 0.022777 4.229 IPS*WPS 2.32 1.918 0.251942 4.222 IP*MSIR 8.50 1.961 0.001186 4.315 IP*WPS 1.38 1.985 0.500172 4.369 MSIR*WPS 6.20 1.945 0.008681 4.282

Confidence level D 95%

4 RESULT AND DISCUSSIONS

There are 26 scenarios for optimizing the four parameters in Table 2 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 5

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 3 Consequently, the polynomial model describing the correlation between overall response and the variables can be rewritten as:

NPVD 244:79 24:63X1C 29:38X2C 15:14X3 0:49X4C 3:4X12 27:68X22

16:85X32 10:77X42C 19:83X1X2 5:08X1X3 2:32X1X4C 8:5X2X3

C 1:38X2X4C 6:2X3X4:

(7)

The analysis of variance, goodness-of-fit, and the adequacy of the model were summarized

in Table 4 The determination coefficient R2 is 0.987 The adjusted determination coefficient R2

AdjD 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.

TABLE 4 ANOVA Analysis

Total 26 1,201,250 46,201.9

Constant 1 1,153,740 1,153,740

Total corrected 25 47,509.3 1,900.37 43.5932 Regression 14 46,882.4 3,348.74 58.7648 0 57.8683 Residual 11 626.841 56.9856 7.54888

N D 26 R2AdjD 0.97

DF D 11 R 2 D 0.987 Cond no D 7.77

Q 2 D 0.902 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 1 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 1 The interaction effect of operating parameters on the NPV.

FIGURE 2 Effects of main factors on the NPV.

Figure 2 indicated that injector-producer spacing of 5 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 5

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 3 and 4, respectively The value of predicted maximum on the surface is confined in the smallest ellipse in the contour diagram