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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
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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

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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
1

Department of Energy and Mineral Resources Engineering, Sejong University,
Seoul, Korea
2
Materials Engineering and Sciences Division, Hongik University, Seoul, Korea
3
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 2 of 0.902 for the NPV response.
Keywords: D-optimal design, net present value, optimization, response surface methodology, steamassisted 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 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:
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ueso.

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APPLICATION OF D-OPTIMAL DESIGN

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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.
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 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

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:
Y D ˇX C ";

(3)


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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:

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ˇ D .X 0 :X/ 1 X 0 Y;

(4)

where X 0 is the transpose of the matrix X, and .X 0 :X/ 1 is the inverse of the matrix X 0 :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

.X 0 X/ 1 ;

(5)

where is the standard deviation, an accurate response surface model is obtained when minimizing
.X 0 X/ 1 . Statistically, minimizing .X 0 X/ 1 is equivalent to maximize the determinant of X 0 X.
This criterion is to generate a design matrix with a maximized jX 0 Xj 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


APPLICATION OF D-OPTIMAL DESIGN

2145

TABLE 1
Reservoir Properties in McMurray Formation, Athabasca

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Parameters
Reservoir size
Reservoir pressure, kPa
Depth to top of reservoir, m
Reservoir thickness, m
Reservoir width, m
Vertical permeability (Kv), Darcy
Permeability ratio (Kh/Kv)
Porosity
Oil saturation
Oil viscosity at reservoir temperature

Reservoir temperature, ı C
Steam quality

McMurray Formation
201 m 900 m
1,500
200
30
201
5
2.5
0.30
0.8
2,000,000
12
0.95

30 m

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 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.


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H. X. NGUYEN ET AL.

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

Coded Level of Variables

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Run
1
2
3
4
5
6
7
8
9
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
1
1
1
1
1
1
1
1
1
1/3
1/3
1/3
1/3
0
0

X2

1
1
1
1
1
1
1
1
1
1
1
1
1/3
1/3
1
1
1
1/3
1/3
1/3
1
1
1
1
0
0

X3

X4

1
1
1
1
1
1
1
1
1/3
1
1
1/3
1
1
1
1/3
1/3
1
1
1
1
1
1
1
0
0

1
1

1
1
1
1
1
1/3
1
1/3
1/3
1
1
1
1/3
1
1
1
1
1
1
1
1
1
0
0

Response
(Simulation Observed)

Actual Level of Variable
IPS, m

IP, kPa

MSIR, m3 /d

WPS, m

Oil Cum., bbl

NPV, $mm

5
17
5
5
17
17
17
5
5
5
5
5
5
5
17
17
17
17
17

17
13
9
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
2
model. The quality of the model is statistically measured by examining R2 , Radj
, and Q2 . The
2

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
2
coefficient Radj
is used to compare the qualities of different models. Any model with values of
2
2
both R and Radj
close to 1 indicates an excellent quality in fitting the observed data. However, it
2
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

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TABLE 3
Regression Coefficients of the Predicted Quadratic Polynomial Model

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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

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:
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 4. The determination coefficient R2 is 0.987. The adjusted determination coefficient

2
.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


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TABLE 4
ANOVA Analysis
NPV

DF

Total
Constant
Total corrected
Regression
Residual

N D 26
DF D 11

26
1
25
14
11

SS
1,201,250
1,153,740
47,509.3
46,882.4
626.841
2
RAdj
D 0.97
R 2 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

0

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 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.


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APPLICATION OF D-OPTIMAL DESIGN

FIGURE 2

2149

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.


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H. X. NGUYEN ET AL.

FIGURE 3 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, 5 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 6.
4.4. Test of Predictive Model
The validity for predicting optimum response is rechecked under an operating condition with IPS
5 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


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APPLICATION OF D-OPTIMAL DESIGN

FIGURE 4

3-D contour plots showing the effects of operation parameters on the NPV.

FIGURE 5

Steam chamber performance in optimal operating condition.

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H. X. NGUYEN ET AL.

FIGURE 6

FIGURE 7

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 2. The result demonstrated the validity of the RSM model,
which is reasonably adequate to predict the performance of SAGD operation (Table 5).

TABLE 5
Observed and Predicted Values under Optimal Conditions

IPS,
m
5

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

7


APPLICATION OF D-OPTIMAL DESIGN

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5. 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 5 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.