SPE-KSA-110188057-MS

Sobol and Halton Sequences: Efficient Experimental Design Techniques towards Improved Uncertainty Quantification of Petroleum ReservoirsMohamed Shams, Amal Petroleum Company, Ahmed H. El-Banbi, and Helmy Sayyouh, Cairo University

Copyright 2017, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition held in Dammam, Saudi Arabia, 24–27 April 2017.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writ -ten consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

AbstractReservoir simulation models often suffer from significant uncertainties due to the lack and inaccuracies of the measured data. Hence, uncertainty analysis and quantification are considered as major concerns in building robust and predictive simulation models. The conventional approach of the uncertainty analysis is conducted by running a very large number of simulation runs in an attempt to capture all effects of the uncertain parameters. Running a large number of simulation runs is costly and very time consuming and therefore efficient approaches have been arisen. One of these efficient approaches is the Experimental Design technique. The experimental design technique is widely applied in different engineering practices for assessing and quantifying uncertainties. Experimental design is the technique used to guide the selection of the samples within the design search space of the uncertain parameters in order to obtain the maximum amount of information using low number of experiments. Several experimental design techniques are applied in petroleum industry and specially in reservoir simulation assisted history matching. Some experimental design techniques shows more effectiveness than others.The objective of this paper is to introduce two efficient experimental design techniques (Halton and Sobol sequences) to the reservoir simulation assisted history matching workflow. This work is to complete the work done by the authors and published in Shams et al. (2017). Shams et al. (2017) applied and tested the potentiality of the two proposed experimental design techniques through a comparative study between their performance in solving assisted history matching problems of different scale material balance problems and the performance of the most widely used experimental design technique, Latin hypercube. In this paper the comparative study is conducted using numerical reservoir simulation model. A performance indicator is developed to compare between the three studied techniques. The performance indicator represents the relative error between the estimated values of history matching parameters calculated using the studied experimental design methods and their exact solutions.The results of this work validate the previous obtained conclusions and indicate that the Sobol and Halton sequences experimental design techniques are more efficient and superior to Latin hypercube method.

IntroductionQuantifying the uncertainties of reservoir simulation models has a significant enhancing impact on the output of the study. Experimental design technique has been considered as one of the most effective approaches used for quantifying the uncertainties of the studied parameters within their search space. Although the experimental design technique was first proposed by Ronald Fisher (1925) for agricultural

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applications, it was first applied in reservoir engineering practices by Damsleth et al. (1992). Damsleth et al. (1992) used experimental design technique to perform a sensitivity analysis study for a field in the North Sea with minimum number of simulation runs. They stated that applying experimental design approach helped to reduce the number of required simulation runs by 30-40% comparing with the procedure that varies one parameter at a time.One of the most widely application area of experimental design in petroleum engineering is the reservoir simulation assisted history matching. In assisted history matching, experimental design concept is associated with providing proxy models that are optimized to give the values of the history matching uncertain parameters. Watkins and Parish (1992a), Watkins and Parish (1992b), Parish et al. (1993), Parish and Little (1994), Eide et al. (1994), Parish and Little (1997), Craig et al. (1997), Van Elk et al. (2000), Venkataraman (2000), Friedmann et al. (2003), Yeten et al. (2005), and Tipping et al. (2008) applied different experimental design methods in reservoir simulation assisted history matching. Van Der Corput (1935), Plackett and Burman (1946), Box and Behnken (1960), Halton sequence (1960), and Sobol sequence (1967) introduced different concepts of experimental design. This work focuses on introducing both Halton (1960) and Sobol (1967) sequences techniques to the area of reservoir simulation assisted history matching as they have not been introduced before. To show the superiority and efficiency of the Halton and Sobol sequences design techniques, a comparative study is conducted to compare between their performances in solving reservoir simulation assisted history matching problems with the performance of the most widely used experimental design technique, Latin hypercube.Latin hypercube, Halton, and Sobol sequences are individuals of one experimental design family called space filling design. Space filling design concept is based on spreading the samples around the design search space without following a specific factors leveling scheme. The number of selected samples is predefined by the designer and doesn't depend on the number of the problem factors or their levels. In Latin hypercube sampling technique, samples are selected according to a uniform distribution as indicated in Fig. 1. Each vertical and horizontal strata contains single sample whereas the location of the sample is still random in the corresponding small square.

Fig. 1: Latin hypercube sampling scheme

Both Halton and Sobol sequences experimental design techniques provide better distribution of the samples than the Latin hypercube technique for a given sample size and parameter space dimensionality. In case of Halton and Sobol sequences, successive samples are located in positions that are as far as possible from existing samples and as a consequence clustering and holes can be avoided. Halton

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sequence method uses base-two Van Der Corput sequence for the first dimension, base-three sequence in the second dimension, base-five in the third dimension, and so on. Sobol sequence method uses only one base for all dimensions and a different permutation of the vector elements for each dimension. Fig. 2 shows a graphical comparison between Latin hypercube, Halton, and Sobol sequences experimental design techniques for a thousand samples in two dimensional search spaces. As indicated in Fig. 2, it is clear that the Sobol sequence method gives the most uniformly distributed samples.

ʘ Latin hypercube design ʘ Halton sequence design ʘSobol sequence design

Fig. 2: A comparison between Latin hypercube, Halton, and Sobol sequences experimental design techniques for 1,000 samples in two dimensional search spaces

MethodologyKnown-solution reservoir simulation history matching problem is used to compare between the performances of the Sobol and Halton sequences techniques and the Latin hypercube. MATLAB® open source codes of the three experimental design methods were used in our work. The number of selected samples and the search space of the uncertain parameters are kept the same to be fair in comparing the different algorithms. Gullfaks published simulation model [Talukdar (2008)] is used to test the three techniques and obtain the results. The assisted history matching test problem is constructed according to the following strategy:

1. Use ECLIPSE® and Petrel® reservoir simulation software package to predict pressure and production performances using known reservoir parameters for 10 years.

2. Use the predicted pressure and production data as a historical data in the test problem.3. Alter the original reservoir parameter to obtain unmatched pressure and production data

performances.4. Use Latin hypercube, Sobol sequence, and Halton sequence experimental design techniques to

select 25 samples of the history matching parameters. These selected samples are used to build the scoping runs.

5. Run the scoping runs and calculate the objective function using the following formula;

f ( x )=∑nw

1n

{∑i

¿¿¿¿

Where n is the number of the observed data points, yi*(x) represents the simulated data obtained using the set of parameters x’s and yi are the observed data, and wi are the weights assigned to each set of data in the function.

6. Build proxy model that interpolates the objective function with the experimentally designed reservoir uncertain parameters samples. For fair comparison among the experimental design techniques, we built all proxy models using Artificial Neural Network technique, ANN.

7. Minimize the created ANN proxy model using genetic optimization algorithm. This minimization process is run five times to take a more representative solution. Consistency was maintained in running the optimization algorithm for the three tested experimental design techniques to make sure of a fair comparison.

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8. Use the following performance indicator to quantify the error between the estimated and exact solutions of the reservoir uncertain history matching parameters.

Performance Indicator=Avg . [|.|( Estimated solution−Exact solutionExact solution

x100)]Test ProblemGullfaks reservoir simulation model, Fig. 3, is used to test the potential of Halton and Sobol sequence techniques and compare their performances with Latin hypercube technique. The model dimensions are (50x49x199) in I, J, and K directions respectively with 487,750 total number of cells.

Fig. 3: Gullfaks 3D simulation model

Fig. 4 shows the faulting system of the model consisting of 17 faults. Only two faults, F3 and F4, are completely sealing faults and as a consequence the model is compartmentalized into three fault blocks; region 1, region 2, and region 3 as shown in Fig. 5. The three compartments have different initial OWC’s and GOC’s.

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Fig. 4: Faulting system of Gullfaks simulation model

Fig. 5: Gullfaks model compartmentalization

Fig. 6 shows some grid properties and their distribution over the static model. Two rock types are encountered in the model and each rock type has its own relative permeability and capillary pressure data.

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Fig. 6: Gullfaks model grid properties

A North East aquifer is modeled analytically by a Carter Tracy aquifer model as presented in Fig. 7. The model contains twenty three wells, twenty producers and three injectors.

Fig. 7: North East Carter Tracy aquifer model

Fig. 8 shows the pressure and production data performances on field basis.

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Fig. 8: Historical field pressure and production data performances

To apply the assisted history matching procedure, fifty history matching parameters are selected. For each history matching parameter, an uncertainty range and a most likely (ML) value are assigned as shown in Table 1.

Table 1: Selected history matching parameters and their uncertainty range

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Min. Max.

1 Region 1_OWC, m X1 -2407 -2403 -2397 -2409

2 Region 2_OWC, m X2 -2406 -2402 -2396 -2408

3 Region 3_OWC, m X3 -2462 -2457 -2451 -2463

4 Region 1_GOC, m X4 -2360 -2359 -2352 -2365

5 Region 2_GOC, m X5 -2333 -2331 -2325 -2337

6 Region 3_GOC, m X6 -2376 -2375 -2372 -2378

7 Perm X,Y_Multiplier X7 1 1.25 0.5 2

8 Perm Z_Multiplier X8 1 5 0.1 10

9 Aquifer Permearbility, md X9 200 500 100 900

10 Aquifer Porosity, fr X10 0.25 0.225 0.15 0.3

11 Aquifer Thickness, m X11 15.24 13.716 6 21

12 Aquifer Angle of Influence, degree X12 360 270 180 380

13 Aquifer External Radius, m X13 152.4 182.88 61 305

14 Fault 1_Transmissibility Multiplier X14 0.5 0.35 0.1 0.6

15 Fault 5_Transmissibility Multiplier X15 0.9 0.75 0.5 1

16 Fault 6_Transmissibility Multiplier X16 0.7 0.6 0.5 0.7

17 Fault 8_Transmissibility Multiplier X17 0.8 0.75 0.5 1

18 Fault 9_Transmissibility Multiplier X18 0.25 0.15 0 0.3

19 Fault 11_Transmissibility Multiplier X19 0.45 0.5 0.4 0.6

20 Fault 13_Transmissibility Multiplier X20 0.6 0.5 0.4 0.6

21 Fault 14_Transmissibility Multiplier X21 0.1 0.075 0 0.15

22 Fault 15_Transmissibility Multiplier X22 0.45 0.4 0.3 0.5

23 Fault 16_Transmissibility Multiplier X23 0.8 0.75 0.5 1

24 RT1_Critical Water Saturation, fr X24 0.35 0.335 0.3 0.37

25 RT1_Corey Exponet of Water (Krow) X25 4 5.5 4 7

26 RT1_Krw@Sorw, fr X26 0.7 0.7 0.65 0.75

27 RT1_Corey Exponet of Oil (Krow) X27 3 3.5 2 5

28 RT1_Corey Exponet of Oil (Krog) X28 3 4 3 5

29 RT1_Kro@Somax, fr X29 0.8 0.775 0.7 0.85

30 RT1_Corey Exponet of Gas (Krog) X30 6 5 4 6

31 RT1_Krg@Swmin, fr X31 0.8 0.85 0.8 0.9

32 RT1_Krg@Sorg, fr X32 0.7 0.7 0.6 0.75

33 RT2_Critical Water Saturation, fr X33 0.22 0.25 0.2 0.3

34 RT2_Corey Exponet of Water (Krow) X34 4 4.5 3 6

35 RT2_Krw@Sorw, fr X35 0.8 0.725 0.6 0.85

36 RT2_Corey Exponet of Oil (Krow) X36 3 4 2 6

37 RT2_Corey Exponet of Oil (Krog) X37 3 4.5 3 6

38 RT2_Kro@Somax, fr X38 0.9 0.75 0.6 0.9

39 RT2_Corey Exponet of Gas (Krog) X39 6 4 2 6

40 RT2_Krg@Swmin, fr X40 0.9 0.85 0.82 0.95

41 RT2_Krg@Sorg, fr X41 0.8 0.7 0.6 0.8

42 Well P402 Skin X42 3 4 2 6

43 Well P404 Skin X43 2 3 1 4

44 Well P408 Skin X44 6 8 1 7

45 Well P409 Skin X45 4 4 3 5

46 Well P501 Skin X46 5 3.5 2 7

47 Well P503 Skin X47 7 4 1 7

48 Well P504 Skin X48 4 5 4 6

49 Well P506 Skin X49 4 3.5 2 5

50 Well P701 Skin X50 2 2.5 1 4

Uncertainity RangeNumber HM Parameter Symbol Exact

SolutionML

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Tables 2, 3 and 4 show the 26 samples selected by the three studied experimental design techniques, Latin hypercube, Sobol sequence, and Halton sequence respectively (sample number 26 represents the most likely values of the history matching parameters, ML). Each experiment represents the input data for a scoping run. For each scoping run, the objective function is calculated.

Table 2: Latin hypercube samples and corresponding calculated objective functionX1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35 X36 X37 X38 X39 X40 X41 X42 X43 X44 X45 X46 X47 X48 X49 X50

-2399.50 -2402.50 -2461.00 -2359.58 -2327.50 -2373.25 1.50 4.23 400.00 0.19 16.89 288.33 294.64 0.58 0.58 0.50 0.71 0.21 0.56 0.51 0.05 0.38 0.94 0.33 6.00 0.71 4.25 3.00 0.78 5.25 0.87 0.61 0.23 3.25 0.81 3.67 5.50 0.71 2.17 0.92 0.63 2.33 1.00 4.00 3.25 5.13 5.50 4.92 4.25 3.63 9.76-2407.50 -2396.50 -2462.50 -2358.50 -2336.00 -2372.25 1.44 2.99 366.67 0.27 13.08 330.00 223.52 0.54 0.50 0.54 0.65 0.10 0.52 0.53 0.01 0.36 0.90 0.31 6.38 0.68 3.00 3.42 0.73 4.42 0.87 0.62 0.30 5.00 0.79 5.17 4.00 0.61 3.33 0.91 0.77 5.83 3.75 5.75 3.33 4.92 6.00 4.67 4.75 3.88 9.80-2408.00 -2399.00 -2462.00 -2353.63 -2334.50 -2374.25 1.19 2.57 666.67 0.29 6.73 321.67 284.48 0.48 0.69 0.68 0.54 0.26 0.40 0.54 0.04 0.45 0.81 0.32 6.63 0.67 2.75 4.17 0.79 5.75 0.86 0.61 0.25 5.75 0.76 5.33 4.75 0.73 5.83 0.87 0.64 3.50 3.25 1.25 4.42 5.75 6.75 5.42 3.50 3.25 8.36-2404.00 -2403.00 -2453.00 -2353.08 -2337.00 -2376.25 0.62 5.05 633.33 0.22 17.53 205.00 152.40 0.22 0.77 0.55 0.88 0.11 0.48 0.41 0.09 0.31 0.75 0.34 4.13 0.72 3.75 3.25 0.73 5.58 0.81 0.64 0.25 4.38 0.67 3.83 5.88 0.83 4.67 0.91 0.78 3.33 1.38 6.00 4.75 4.29 4.50 5.17 4.50 1.13 9.02-2408.50 -2398.50 -2457.00 -2361.21 -2336.50 -2375.50 0.50 8.35 233.33 0.21 14.35 213.33 121.92 0.35 0.65 0.58 0.98 0.09 0.46 0.59 0.11 0.49 0.83 0.36 4.00 0.73 2.00 3.08 0.84 5.67 0.82 0.70 0.29 4.63 0.64 2.17 3.13 0.89 5.00 0.84 0.73 3.17 3.38 5.00 3.92 2.63 3.75 4.25 2.75 2.00 10.55-2401.50 -2406.50 -2456.50 -2354.17 -2331.50 -2377.50 0.81 7.94 166.67 0.21 18.16 305.00 81.28 0.56 0.92 0.60 0.92 0.29 0.44 0.55 0.02 0.37 0.69 0.34 4.38 0.70 2.25 4.25 0.82 5.33 0.81 0.60 0.28 5.88 0.73 5.00 5.00 0.70 4.33 0.84 0.60 2.17 2.63 3.25 3.08 3.46 5.25 4.58 4.63 1.50 8.55-2404.50 -2407.50 -2463.00 -2362.83 -2335.50 -2373.75 1.75 9.59 133.33 0.19 8.00 255.00 254.00 0.18 0.96 0.53 0.96 0.05 0.45 0.43 0.02 0.44 0.85 0.34 5.13 0.66 3.63 3.33 0.71 4.92 0.80 0.63 0.28 3.63 0.84 5.50 4.25 0.84 3.00 0.90 0.78 2.50 1.25 5.25 4.08 6.58 6.50 4.33 3.75 2.25 7.39-2403.50 -2400.00 -2455.00 -2352.54 -2328.50 -2377.25 1.31 0.51 500.00 0.18 11.18 355.00 264.16 0.37 0.88 0.63 0.73 0.01 0.49 0.48 0.01 0.38 0.50 0.31 5.00 0.69 2.50 4.75 0.84 4.50 0.82 0.63 0.21 3.75 0.74 4.83 3.25 0.86 3.67 0.82 0.70 4.83 3.13 3.50 4.50 3.25 6.25 5.08 4.38 1.88 10.44-2405.50 -2403.50 -2460.00 -2357.42 -2333.50 -2376.75 1.88 2.16 266.67 0.15 7.37 188.33 182.88 0.52 0.85 0.52 0.81 0.06 0.43 0.53 0.08 0.33 0.88 0.37 5.63 0.65 2.38 3.58 0.77 4.08 0.85 0.67 0.22 4.88 0.61 2.00 4.38 0.80 2.67 0.93 0.65 3.67 3.00 4.75 3.75 4.08 5.75 4.08 3.00 2.75 7.11-2400.00 -2399.50 -2458.50 -2361.75 -2333.00 -2372.50 1.56 5.87 100.00 0.17 14.99 371.67 111.76 0.27 0.73 0.62 0.63 0.25 0.58 0.43 0.04 0.33 0.56 0.32 5.38 0.75 3.38 3.92 0.81 5.17 0.84 0.65 0.26 5.38 0.66 4.33 3.63 0.68 2.50 0.90 0.72 5.17 2.00 5.50 4.25 2.00 3.00 4.42 2.88 3.50 9.45-2409.00 -2397.50 -2455.50 -2362.29 -2330.50 -2375.75 0.75 6.29 533.33 0.28 16.26 313.33 162.56 0.43 0.54 0.53 0.58 0.14 0.47 0.49 0.06 0.48 0.96 0.36 4.63 0.67 4.00 4.08 0.79 5.92 0.80 0.69 0.26 3.38 0.82 4.67 3.50 0.85 4.00 0.89 0.71 2.83 2.13 6.75 3.58 2.83 5.00 5.75 3.25 3.13 11.58-2400.50 -2400.50 -2454.00 -2356.88 -2330.00 -2375.00 1.06 0.92 200.00 0.24 6.10 271.67 213.36 0.39 0.67 0.56 0.69 0.16 0.42 0.58 0.00 0.39 0.77 0.31 4.75 0.73 4.75 3.50 0.80 5.08 0.85 0.68 0.27 4.75 0.69 4.50 5.13 0.75 5.50 0.94 0.67 3.83 2.50 1.50 4.92 2.42 3.50 5.92 4.88 2.50 6.86-2403.00 -2402.00 -2457.00 -2358.50 -2331.00 -2375.00 1.25 5.05 500.00 0.22 13.72 280.00 182.88 0.35 0.75 0.60 0.75 0.15 0.50 0.50 0.08 0.40 0.75 0.33 5.50 0.70 3.50 4.00 0.77 5.00 0.85 0.68 0.25 4.50 0.73 4.00 4.50 0.75 4.00 0.89 0.70 4.00 2.50 4.00 4.00 4.50 4.00 5.00 3.50 2.50 9.10-2406.00 -2404.00 -2460.50 -2360.67 -2332.50 -2375.25 1.38 8.76 766.67 0.20 20.70 280.00 142.24 0.29 0.81 0.63 0.79 0.12 0.58 0.42 0.14 0.40 0.71 0.36 6.13 0.66 2.13 4.42 0.74 4.83 0.84 0.66 0.23 4.13 0.75 3.33 3.75 0.74 2.33 0.83 0.73 4.00 2.38 6.25 3.00 6.38 4.25 4.00 2.00 2.38 7.96-2397.50 -2407.00 -2459.00 -2355.25 -2332.00 -2374.50 1.69 3.40 433.33 0.16 10.54 238.33 193.04 0.25 0.94 0.66 0.90 0.15 0.53 0.50 0.08 0.32 0.52 0.30 6.25 0.72 2.88 3.83 0.75 4.00 0.90 0.66 0.24 5.50 0.62 3.17 5.38 0.64 3.83 0.88 0.68 5.00 2.75 1.00 4.33 5.96 2.75 4.17 3.63 1.75 7.38-2406.50 -2405.00 -2456.00 -2355.79 -2329.50 -2377.75 0.88 3.81 866.67 0.28 11.81 180.00 243.84 0.41 0.75 0.58 0.85 0.04 0.41 0.57 0.11 0.47 0.92 0.35 5.50 0.65 3.50 4.00 0.74 4.75 0.86 0.69 0.23 3.50 0.78 3.50 5.25 0.81 5.33 0.87 0.68 2.67 2.88 2.25 3.67 6.79 4.75 5.50 4.00 1.38 7.88-2401.00 -2401.00 -2454.50 -2360.13 -2329.00 -2373.50 0.56 7.53 700.00 0.29 19.43 363.33 172.72 0.16 0.63 0.67 0.67 0.22 0.57 0.47 0.07 0.46 0.60 0.30 5.25 0.74 4.50 4.33 0.77 5.83 0.85 0.68 0.28 4.00 0.83 5.83 4.50 0.69 5.17 0.83 0.74 4.17 1.88 3.00 4.17 4.71 2.00 5.83 3.88 2.13 10.04-2403.00 -2404.50 -2459.50 -2365.00 -2334.00 -2373.00 1.13 9.18 466.67 0.26 15.62 196.67 91.44 0.31 0.60 0.57 0.75 0.27 0.50 0.52 0.13 0.42 0.98 0.36 5.88 0.70 4.38 3.17 0.70 5.42 0.88 0.71 0.29 5.13 0.70 3.00 5.63 0.63 4.17 0.94 0.69 3.00 1.75 4.25 3.42 5.54 1.50 4.83 2.50 3.00 7.55-2402.00 -2397.00 -2451.50 -2354.71 -2327.00 -2376.50 0.69 0.10 833.33 0.26 18.80 296.67 101.60 0.50 0.52 0.67 0.52 0.24 0.54 0.56 0.12 0.35 0.63 0.33 5.75 0.74 3.25 4.58 0.83 5.00 0.89 0.72 0.22 5.25 0.60 2.33 4.63 0.65 4.83 0.86 0.62 5.33 3.63 2.50 3.83 2.21 1.25 5.58 3.13 2.63 9.03-2405.00 -2398.00 -2458.00 -2363.37 -2331.00 -2372.75 1.63 1.75 800.00 0.24 8.64 246.67 274.32 0.12 0.56 0.59 0.56 0.00 0.55 0.44 0.12 0.43 0.79 0.33 6.50 0.70 4.63 3.67 0.72 4.58 0.89 0.74 0.21 3.00 0.71 2.83 3.88 0.79 3.50 0.93 0.79 5.67 2.25 4.50 4.83 5.33 2.50 5.33 2.25 3.38 11.06-2398.00 -2406.00 -2457.50 -2356.33 -2326.00 -2374.75 1.94 1.34 733.33 0.23 12.45 338.33 233.68 0.33 0.83 0.61 0.50 0.20 0.53 0.40 0.03 0.30 0.65 0.31 6.88 0.67 4.88 4.83 0.71 4.25 0.88 0.64 0.20 4.25 0.80 5.67 5.75 0.60 2.83 0.92 0.66 4.67 1.50 2.75 4.00 6.17 4.00 5.67 4.13 2.88 8.68-2402.50 -2401.50 -2461.50 -2364.46 -2325.50 -2374.00 1.81 4.64 333.33 0.22 9.27 346.67 203.20 0.45 0.71 0.64 0.60 0.17 0.51 0.58 0.06 0.48 0.73 0.32 6.75 0.68 3.13 4.67 0.81 4.33 0.88 0.71 0.24 4.50 0.77 4.00 3.00 0.66 3.17 0.85 0.61 4.50 3.50 1.75 3.17 4.50 3.25 4.75 2.38 3.75 8.89-2398.50 -2405.50 -2452.50 -2363.92 -2328.00 -2376.00 1.25 7.11 300.00 0.16 20.07 230.00 71.12 0.20 0.79 0.51 0.94 0.02 0.59 0.45 0.10 0.34 0.67 0.35 4.25 0.73 4.13 3.75 0.76 4.17 0.83 0.73 0.25 3.13 0.68 2.50 4.13 0.76 2.00 0.89 0.75 4.33 1.63 6.50 3.50 3.04 1.00 4.50 3.38 1.63 7.95-2399.00 -2408.00 -2452.00 -2359.04 -2326.50 -2378.00 1.00 6.70 600.00 0.18 13.72 221.67 132.08 0.14 0.98 0.65 0.83 0.19 0.48 0.46 0.14 0.43 0.58 0.35 4.50 0.71 3.88 4.50 0.82 5.50 0.83 0.73 0.20 3.88 0.65 2.67 4.88 0.88 4.50 0.86 0.63 2.00 1.13 2.00 4.58 3.88 1.75 5.25 2.13 1.00 6.29-2407.00 -2402.00 -2453.50 -2357.96 -2335.00 -2377.00 0.94 5.46 566.67 0.25 9.91 263.33 60.96 0.10 0.90 0.69 0.77 0.08 0.43 0.48 0.09 0.41 0.54 0.33 4.88 0.69 2.63 4.92 0.76 4.67 0.83 0.74 0.27 5.63 0.63 4.17 3.38 0.78 5.67 0.85 0.76 5.50 3.88 3.75 4.67 3.67 2.25 5.00 2.63 1.25 7.65-2403.35 -2401.82 -2456.69 -2358.54 -2331.11 -2375.00 1.25 5.00 500.00 0.23 13.72 270.00 182.88 0.35 0.75 0.60 0.75 0.15 0.50 0.50 0.08 0.40 0.75 0.34 5.50 0.70 3.50 4.00 0.78 5.00 0.70 0.70 0.25 4.50 0.73 4.00 4.50 0.75 4.00 0.85 0.70 4.00 3.00 8.00 4.00 3.50 4.00 5.00 3.50 2.50 9.16

Obj. FnExperimentNumber

876

1617

242526

181920212223

54321

12131415

91011

Table 3: Sobol Sequence samples and corresponding calculated objective functionX1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35 X36 X37 X38 X39 X40 X41 X42 X43 X44 X45 X46 X47 X48 X49 X50

-2409.44 -2407.92 -2462.78 -2364.64 -2337.21 -2378.05 0.50 0.10 100.00 0.15 6.10 180.00 60.96 0.10 0.50 0.50 0.50 0.00 0.40 0.40 0.00 0.30 0.50 0.30 4.00 0.65 2.00 3.00 0.70 4.00 0.80 0.60 0.20 3.00 0.60 2.00 3.00 0.60 2.00 0.82 0.60 2.00 1.00 1.00 3.00 2.00 1.00 4.00 2.00 1.00 10.73-2403.35 -2401.82 -2456.69 -2358.54 -2331.11 -2375.00 1.25 5.05 500.00 0.22 13.72 280.00 182.88 0.35 0.75 0.60 0.75 0.15 0.50 0.50 0.08 0.40 0.75 0.33 5.50 0.70 3.50 4.00 0.77 5.00 0.85 0.68 0.25 4.50 0.73 4.00 4.50 0.75 4.00 0.89 0.70 4.00 2.50 4.00 4.00 4.50 4.00 5.00 3.50 2.50 9.18-2406.40 -2398.78 -2459.74 -2355.49 -2334.16 -2373.48 0.88 7.53 700.00 0.19 17.53 230.00 243.84 0.22 0.88 0.55 0.63 0.22 0.45 0.55 0.04 0.45 0.63 0.35 6.25 0.67 2.75 3.50 0.81 4.50 0.88 0.64 0.28 5.25 0.66 5.00 3.75 0.68 3.00 0.92 0.65 5.00 1.75 2.50 3.50 5.75 5.50 4.50 4.25 3.25 10.83-2400.30 -2404.87 -2453.64 -2361.59 -2328.06 -2376.53 1.63 2.57 300.00 0.26 9.91 330.00 121.92 0.48 0.63 0.65 0.88 0.08 0.55 0.45 0.11 0.35 0.88 0.32 4.75 0.73 4.25 4.50 0.74 5.50 0.83 0.71 0.23 3.75 0.79 3.00 5.25 0.83 5.00 0.85 0.75 3.00 3.25 5.50 4.50 3.25 2.50 5.50 2.75 1.75 7.28-2407.92 -2400.30 -2452.12 -2353.97 -2329.59 -2377.29 1.06 3.81 800.00 0.24 15.62 355.00 274.32 0.16 0.69 0.58 0.94 0.19 0.43 0.43 0.09 0.38 0.94 0.31 6.63 0.71 2.38 3.75 0.83 5.75 0.81 0.62 0.21 4.88 0.82 5.50 4.88 0.64 3.50 0.87 0.78 4.50 2.88 4.75 3.75 3.88 3.25 4.25 2.38 2.88 11.07-2401.82 -2406.40 -2458.21 -2360.07 -2335.68 -2374.24 1.81 8.76 400.00 0.17 8.00 255.00 152.40 0.41 0.94 0.67 0.69 0.04 0.53 0.53 0.02 0.48 0.69 0.34 5.13 0.66 3.88 4.75 0.76 4.75 0.86 0.69 0.26 3.38 0.69 3.50 3.38 0.79 5.50 0.93 0.68 2.50 1.38 1.75 4.75 6.38 6.25 5.25 3.88 1.38 5.24-2404.87 -2403.35 -2455.16 -2363.11 -2326.54 -2372.72 0.69 6.29 200.00 0.28 11.81 305.00 91.44 0.29 0.81 0.53 0.81 0.11 0.48 0.58 0.13 0.43 0.81 0.36 4.38 0.74 3.13 3.25 0.72 5.25 0.89 0.66 0.29 4.13 0.76 2.50 5.63 0.71 2.50 0.90 0.73 3.50 3.63 6.25 3.25 5.13 4.75 4.75 4.63 2.13 8.47-2398.78 -2397.25 -2461.26 -2357.02 -2332.63 -2375.76 1.44 1.34 600.00 0.21 19.43 205.00 213.36 0.54 0.56 0.63 0.56 0.26 0.58 0.48 0.06 0.33 0.56 0.33 5.88 0.69 4.63 4.25 0.79 4.25 0.84 0.73 0.24 5.63 0.63 4.50 4.13 0.86 4.50 0.84 0.63 5.50 2.13 3.25 4.25 2.63 1.75 5.75 3.13 3.63 10.07-2408.68 -2396.49 -2454.40 -2359.30 -2327.30 -2377.67 1.16 5.67 750.00 0.25 7.05 217.50 167.64 0.38 0.66 0.66 0.91 0.21 0.44 0.59 0.05 0.34 0.97 0.33 5.69 0.73 3.69 3.13 0.80 4.88 0.83 0.74 0.21 5.81 0.77 3.25 5.06 0.62 3.75 0.89 0.69 3.75 1.19 6.63 4.88 6.69 5.88 4.38 2.56 3.81 11.18-2402.59 -2402.59 -2460.50 -2353.21 -2333.40 -2374.62 1.91 0.72 350.00 0.18 14.67 317.50 289.56 0.13 0.91 0.56 0.66 0.06 0.54 0.49 0.12 0.44 0.72 0.37 4.19 0.68 2.19 4.13 0.73 5.88 0.88 0.67 0.26 4.31 0.65 5.25 3.56 0.77 5.75 0.83 0.79 5.75 2.69 3.63 3.88 4.19 2.88 5.38 4.06 2.31 5.71-2405.63 -2405.63 -2451.35 -2356.26 -2330.35 -2373.10 0.78 3.19 150.00 0.29 18.48 267.50 228.60 0.51 0.78 0.61 0.78 0.13 0.49 0.44 0.01 0.49 0.84 0.35 4.94 0.71 4.44 3.63 0.77 4.38 0.86 0.70 0.28 3.56 0.83 4.25 5.81 0.69 2.75 0.86 0.64 4.75 1.94 5.13 4.38 2.94 1.38 4.88 4.81 1.56 8.47-2399.54 -2399.54 -2457.45 -2362.35 -2336.44 -2376.14 1.53 8.14 550.00 0.22 10.86 367.50 106.68 0.26 0.53 0.51 0.53 0.28 0.59 0.54 0.08 0.39 0.59 0.31 6.44 0.66 2.94 4.63 0.84 5.38 0.81 0.63 0.23 5.06 0.71 2.25 4.31 0.84 4.75 0.93 0.74 2.75 3.44 2.13 3.38 5.44 4.38 5.88 3.31 3.06 9.37-2407.16 -2404.11 -2458.97 -2357.78 -2331.87 -2376.91 0.59 9.38 250.00 0.16 16.57 342.50 198.12 0.44 0.59 0.64 0.59 0.02 0.41 0.56 0.14 0.36 0.53 0.32 5.31 0.69 4.06 3.88 0.75 5.13 0.84 0.72 0.22 3.94 0.68 4.75 3.19 0.66 2.25 0.94 0.71 5.25 3.06 2.88 4.13 4.81 5.13 4.13 2.19 1.94 9.03-2401.06 -2398.01 -2452.88 -2363.88 -2325.78 -2373.86 1.34 4.43 650.00 0.23 8.95 242.50 76.20 0.19 0.84 0.54 0.84 0.17 0.51 0.46 0.07 0.46 0.78 0.36 6.81 0.74 2.56 4.88 0.82 4.13 0.89 0.65 0.27 5.44 0.80 2.75 4.69 0.81 4.25 0.88 0.61 3.25 1.56 5.88 3.13 2.31 2.13 5.13 3.69 3.44 8.99-2404.11 -2401.06 -2462.02 -2360.83 -2334.92 -2372.33 0.97 1.96 850.00 0.20 12.76 292.50 137.16 0.57 0.97 0.69 0.72 0.24 0.46 0.41 0.10 0.41 0.66 0.34 6.06 0.67 4.81 3.38 0.78 5.63 0.87 0.68 0.29 4.69 0.62 3.75 3.94 0.73 3.25 0.84 0.76 2.25 3.81 1.38 4.63 3.56 3.63 4.63 4.44 2.69 9.88-2398.01 -2407.16 -2455.93 -2354.73 -2328.82 -2375.38 1.72 6.91 450.00 0.27 20.38 192.50 259.08 0.32 0.72 0.59 0.97 0.09 0.56 0.51 0.03 0.31 0.91 0.30 4.56 0.72 3.31 4.38 0.71 4.63 0.82 0.61 0.24 3.19 0.74 5.75 5.44 0.88 5.25 0.91 0.66 4.25 2.31 4.38 3.63 6.06 6.63 5.63 2.94 1.19 7.25-2409.06 -2401.44 -2457.83 -2356.64 -2326.16 -2374.43 1.67 2.88 175.00 0.28 13.24 361.25 220.98 0.46 0.80 0.57 0.89 0.07 0.48 0.51 0.00 0.46 0.95 0.31 6.91 0.68 2.47 4.44 0.83 5.19 0.87 0.62 0.20 4.59 0.70 2.88 4.41 0.68 5.88 0.92 0.62 2.63 3.16 2.31 3.19 4.66 5.31 4.19 2.28 2.97 7.24-2402.97 -2407.54 -2451.74 -2362.73 -2332.25 -2377.48 0.92 7.83 575.00 0.20 20.86 261.25 99.06 0.21 0.55 0.67 0.64 0.22 0.58 0.41 0.08 0.36 0.70 0.34 5.41 0.73 3.97 3.44 0.75 4.19 0.82 0.70 0.25 3.09 0.83 4.88 5.91 0.83 3.88 0.86 0.72 4.63 1.66 5.31 4.19 2.16 2.31 5.19 3.78 1.47 7.89-2406.02 -2404.49 -2460.88 -2359.69 -2329.21 -2375.95 1.30 5.36 775.00 0.24 17.05 311.25 160.02 0.58 0.67 0.52 0.77 0.29 0.43 0.46 0.04 0.31 0.83 0.36 4.66 0.66 3.22 4.94 0.71 5.69 0.84 0.66 0.28 3.84 0.64 5.88 3.66 0.61 4.88 0.82 0.67 5.63 3.91 3.81 3.69 3.41 3.81 4.69 4.53 2.22 8.06-2399.92 -2398.40 -2454.78 -2353.59 -2335.30 -2372.91 0.55 0.41 375.00 0.16 9.43 211.25 281.94 0.33 0.92 0.62 0.52 0.14 0.53 0.56 0.12 0.41 0.58 0.32 6.16 0.71 4.72 3.94 0.79 4.69 0.89 0.74 0.23 5.34 0.76 3.88 5.16 0.76 2.88 0.89 0.77 3.63 2.41 6.81 4.69 5.91 6.81 5.69 3.03 3.72 12.88-2407.54 -2406.02 -2456.31 -2361.21 -2333.78 -2373.67 1.48 1.65 875.00 0.22 18.95 186.25 129.54 0.40 0.98 0.54 0.58 0.18 0.46 0.53 0.10 0.43 0.52 0.32 4.28 0.75 2.09 4.69 0.73 4.94 0.85 0.60 0.22 3.47 0.73 5.38 5.53 0.72 4.38 0.91 0.79 4.13 1.28 4.56 3.94 6.53 6.06 4.44 2.66 1.09 8.86-2401.44 -2399.92 -2462.40 -2355.11 -2327.68 -2376.72 0.73 6.60 475.00 0.30 11.33 286.25 251.46 0.15 0.73 0.64 0.83 0.03 0.56 0.43 0.02 0.33 0.77 0.35 5.78 0.70 3.59 3.69 0.81 5.94 0.80 0.68 0.27 4.97 0.61 3.38 4.03 0.87 2.38 0.84 0.69 2.13 2.78 1.56 4.94 4.03 3.06 5.44 4.16 2.59 9.66-2404.49 -2396.87 -2453.26 -2358.16 -2336.83 -2375.19 1.86 9.07 275.00 0.18 7.52 236.25 190.50 0.52 0.61 0.59 0.70 0.10 0.41 0.48 0.14 0.38 0.64 0.37 6.53 0.72 2.84 4.19 0.85 4.44 0.83 0.64 0.29 5.72 0.80 2.38 4.78 0.65 5.38 0.87 0.74 3.13 2.03 6.06 3.44 2.78 1.56 4.94 4.91 3.34 9.10-2398.40 -2402.97 -2459.36 -2364.26 -2330.73 -2372.14 1.11 4.12 675.00 0.26 15.14 336.25 68.58 0.27 0.86 0.69 0.95 0.25 0.51 0.58 0.06 0.48 0.89 0.33 5.03 0.67 4.34 3.19 0.77 5.44 0.88 0.72 0.24 4.22 0.67 4.38 3.28 0.80 3.38 0.94 0.64 5.13 3.53 3.06 4.44 5.28 4.56 5.94 3.41 1.84 9.02-2408.30 -2402.21 -2452.50 -2354.35 -2334.54 -2374.81 1.58 8.45 725.00 0.21 12.29 348.75 266.70 0.18 0.89 0.61 0.55 0.16 0.47 0.49 0.05 0.49 0.55 0.33 5.22 0.70 4.16 4.31 0.76 5.81 0.90 0.73 0.21 4.41 0.81 3.63 5.34 0.70 4.13 0.86 0.68 3.38 2.97 5.69 4.81 4.34 3.44 4.31 2.47 2.03 9.20-2403.35 -2401.82 -2456.69 -2358.54 -2331.11 -2375.00 1.25 5.00 500.00 0.23 13.72 270.00 182.88 0.35 0.75 0.60 0.75 0.15 0.50 0.50 0.08 0.40 0.75 0.34 5.50 0.70 3.50 4.00 0.78 5.00 0.70 0.70 0.25 4.50 0.73 4.00 4.50 0.75 4.00 0.85 0.70 4.00 3.00 8.00 4.00 3.50 4.00 5.00 3.50 2.50 9.16

Obj. FnExperiment

2526

161718192021222324

789

101112131415

123456

10 SPE-KSA-8665-MS

Table 4: Halton Sequence samples and corresponding calculated objective functionX1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35 X36 X37 X38 X39 X40 X41 X42 X43 X44 X45 X46 X47 X48 X49 X50

-2403.35 -2403.86 -2460.35 -2362.90 -2336.10 -2377.58 0.59 0.62 134.78 0.16 6.59 185.41 66.91 0.11 0.51 0.50 0.51 0.00 0.40 0.40 0.00 0.30 0.51 0.30 4.03 0.65 2.03 3.02 0.70 4.02 0.80 0.60 0.20 3.02 0.60 2.03 3.02 0.60 2.02 0.80 0.60 2.02 1.02 1.03 3.01 2.03 1.03 4.01 2.01 1.01 11.58-2406.40 -2399.79 -2457.91 -2361.15 -2334.99 -2377.11 0.68 1.14 169.57 0.16 7.08 190.81 72.85 0.12 0.52 0.51 0.52 0.01 0.41 0.41 0.00 0.31 0.51 0.30 4.06 0.65 2.06 3.04 0.70 4.04 0.80 0.60 0.20 3.04 0.60 2.05 3.04 0.60 2.05 0.80 0.60 2.04 1.03 1.06 3.02 2.05 1.06 4.02 2.03 1.03 12.33-2400.30 -2406.57 -2455.47 -2359.41 -2333.88 -2376.64 0.76 1.66 204.35 0.17 7.57 196.22 78.80 0.13 0.53 0.51 0.53 0.01 0.41 0.41 0.01 0.31 0.52 0.30 4.09 0.65 2.09 3.06 0.70 4.05 0.80 0.60 0.20 3.06 0.61 2.08 3.06 0.61 2.07 0.80 0.60 2.07 1.05 1.09 3.03 2.08 1.09 4.03 2.04 1.04 8.09-2407.92 -2402.50 -2453.03 -2357.67 -2332.77 -2376.17 0.85 2.18 239.13 0.17 8.06 201.62 84.75 0.15 0.54 0.52 0.53 0.02 0.41 0.41 0.01 0.31 0.52 0.30 4.12 0.65 2.12 3.07 0.71 4.07 0.80 0.60 0.20 3.09 0.61 2.11 3.08 0.61 2.10 0.80 0.60 2.09 1.06 1.12 3.04 2.10 1.11 4.04 2.05 1.05 9.89-2401.82 -2398.44 -2462.30 -2355.93 -2331.66 -2375.70 0.94 2.71 273.91 0.18 8.55 207.03 90.70 0.16 0.55 0.52 0.54 0.02 0.41 0.41 0.01 0.31 0.53 0.30 4.15 0.65 2.15 3.09 0.71 4.09 0.80 0.61 0.20 3.11 0.61 2.13 3.10 0.61 2.12 0.80 0.61 2.11 1.08 1.16 3.05 2.13 1.14 4.04 2.07 1.07 11.54-2404.87 -2405.21 -2459.86 -2354.19 -2330.56 -2375.24 1.03 3.23 308.70 0.18 9.05 212.43 96.64 0.17 0.56 0.52 0.55 0.03 0.42 0.42 0.01 0.32 0.54 0.30 4.19 0.66 2.17 3.11 0.71 4.11 0.80 0.61 0.20 3.13 0.61 2.16 3.11 0.61 2.14 0.81 0.61 2.13 1.09 1.19 3.06 2.15 1.17 4.05 2.08 1.08 8.74-2398.78 -2401.15 -2457.42 -2364.39 -2329.45 -2374.77 1.12 3.75 343.48 0.19 9.54 217.84 102.59 0.18 0.57 0.53 0.56 0.03 0.42 0.42 0.01 0.32 0.54 0.31 4.22 0.66 2.20 3.13 0.71 4.12 0.81 0.61 0.21 3.15 0.61 2.19 3.13 0.61 2.17 0.81 0.61 2.15 1.11 1.22 3.07 2.18 1.20 4.06 2.09 1.09 10.31-2408.68 -2397.08 -2454.98 -2362.65 -2328.34 -2374.30 1.21 4.27 378.26 0.19 10.03 223.24 108.54 0.19 0.59 0.53 0.57 0.04 0.42 0.42 0.02 0.32 0.55 0.31 4.25 0.66 2.23 3.15 0.71 4.14 0.81 0.61 0.21 3.17 0.61 2.21 3.15 0.61 2.19 0.81 0.61 2.18 1.13 1.25 3.08 2.20 1.23 4.07 2.11 1.10 10.91-2402.59 -2407.47 -2452.54 -2360.91 -2327.23 -2373.83 1.29 4.79 413.04 0.20 10.52 228.65 114.49 0.20 0.60 0.53 0.58 0.04 0.43 0.43 0.02 0.32 0.55 0.31 4.28 0.66 2.26 3.17 0.71 4.16 0.81 0.61 0.21 3.19 0.62 2.24 3.17 0.62 2.22 0.81 0.61 2.20 1.14 1.28 3.09 2.23 1.26 4.08 2.12 1.12 7.58-2405.63 -2403.40 -2461.81 -2359.16 -2326.12 -2373.36 1.38 5.31 447.83 0.20 11.01 234.05 120.43 0.22 0.61 0.54 0.58 0.05 0.43 0.43 0.02 0.33 0.56 0.31 4.31 0.66 2.29 3.19 0.71 4.18 0.81 0.61 0.21 3.22 0.62 2.26 3.19 0.62 2.24 0.81 0.61 2.22 1.16 1.31 3.10 2.25 1.28 4.09 2.13 1.13 9.10-2399.54 -2399.34 -2459.37 -2357.42 -2337.11 -2372.89 1.47 5.83 482.61 0.21 11.50 239.46 126.38 0.23 0.62 0.54 0.59 0.05 0.43 0.43 0.02 0.33 0.57 0.31 4.34 0.66 2.32 3.21 0.72 4.19 0.81 0.61 0.21 3.24 0.62 2.29 3.21 0.62 2.26 0.81 0.61 2.24 1.17 1.34 3.11 2.28 1.31 4.10 2.15 1.14 10.65-2407.16 -2406.11 -2456.93 -2355.68 -2336.00 -2372.42 1.56 6.35 517.39 0.21 12.00 244.86 132.33 0.24 0.63 0.55 0.60 0.06 0.44 0.43 0.02 0.33 0.57 0.31 4.37 0.66 2.35 3.22 0.72 4.21 0.81 0.61 0.21 3.26 0.62 2.32 3.23 0.62 2.29 0.81 0.61 2.27 1.19 1.37 3.12 2.30 1.34 4.11 2.16 1.16 7.90-2401.06 -2402.05 -2454.49 -2353.94 -2334.89 -2378.01 1.65 6.87 552.17 0.22 12.49 250.27 138.28 0.25 0.64 0.55 0.61 0.06 0.44 0.44 0.03 0.33 0.58 0.31 4.40 0.66 2.38 3.24 0.72 4.23 0.81 0.61 0.21 3.28 0.62 2.34 3.25 0.62 2.31 0.81 0.61 2.29 1.20 1.40 3.13 2.33 1.37 4.12 2.17 1.17 9.58-2404.11 -2397.99 -2452.06 -2364.14 -2333.78 -2377.54 1.74 7.39 586.96 0.22 12.98 255.68 144.22 0.26 0.65 0.55 0.62 0.07 0.44 0.44 0.03 0.34 0.58 0.31 4.43 0.66 2.41 3.26 0.72 4.25 0.81 0.62 0.21 3.30 0.62 2.37 3.27 0.63 2.34 0.81 0.62 2.31 1.22 1.44 3.14 2.35 1.40 4.13 2.19 1.18 10.99-2398.01 -2404.76 -2461.32 -2362.40 -2332.67 -2377.08 1.82 7.92 621.74 0.23 13.47 261.08 150.17 0.27 0.66 0.56 0.63 0.07 0.44 0.44 0.03 0.34 0.59 0.31 4.46 0.66 2.44 3.28 0.72 4.27 0.81 0.62 0.21 3.32 0.63 2.40 3.29 0.63 2.36 0.81 0.62 2.33 1.24 1.47 3.15 2.38 1.43 4.13 2.20 1.20 8.09-2409.06 -2400.70 -2458.88 -2360.66 -2331.56 -2376.61 1.91 8.44 656.52 0.23 13.96 266.49 156.12 0.29 0.67 0.56 0.64 0.08 0.45 0.45 0.03 0.34 0.60 0.31 4.49 0.67 2.47 3.30 0.72 4.28 0.81 0.62 0.21 3.35 0.63 2.42 3.31 0.63 2.38 0.81 0.62 2.35 1.25 1.50 3.16 2.40 1.45 4.14 2.21 1.21 9.78-2402.97 -2396.63 -2456.44 -2358.92 -2330.46 -2376.14 0.51 8.96 691.30 0.24 14.45 271.89 162.06 0.30 0.68 0.56 0.64 0.08 0.45 0.45 0.03 0.34 0.60 0.31 4.53 0.67 2.50 3.32 0.72 4.30 0.81 0.62 0.21 3.37 0.63 2.45 3.32 0.63 2.41 0.81 0.62 2.38 1.27 1.53 3.17 2.43 1.48 4.15 2.22 1.22 13.12-2406.02 -2407.02 -2454.01 -2357.17 -2329.35 -2375.67 0.59 9.48 726.09 0.24 14.95 277.30 168.01 0.31 0.69 0.57 0.65 0.09 0.45 0.45 0.04 0.35 0.61 0.31 4.56 0.67 2.52 3.34 0.72 4.32 0.81 0.62 0.21 3.39 0.63 2.48 3.34 0.63 2.43 0.82 0.62 2.40 1.28 1.56 3.18 2.45 1.51 4.16 2.24 1.24 9.24-2399.92 -2402.95 -2451.57 -2355.43 -2328.24 -2375.20 0.68 0.13 760.87 0.25 15.44 282.70 173.96 0.32 0.70 0.57 0.66 0.09 0.46 0.45 0.04 0.35 0.61 0.31 4.59 0.67 2.55 3.36 0.73 4.34 0.81 0.62 0.21 3.41 0.63 2.50 3.36 0.63 2.46 0.82 0.62 2.42 1.30 1.59 3.19 2.48 1.54 4.17 2.25 1.25 11.76-2407.54 -2398.89 -2460.83 -2353.69 -2327.13 -2374.73 0.77 0.65 795.65 0.25 15.93 288.11 179.91 0.33 0.71 0.58 0.67 0.10 0.46 0.46 0.04 0.35 0.62 0.32 4.62 0.67 2.58 3.37 0.73 4.35 0.82 0.62 0.21 3.43 0.63 2.53 3.38 0.64 2.48 0.82 0.62 2.44 1.31 1.62 3.20 2.50 1.57 4.18 2.26 1.26 12.41-2401.44 -2405.66 -2458.39 -2363.89 -2326.02 -2374.26 0.86 1.17 830.43 0.26 16.42 293.51 185.85 0.34 0.72 0.58 0.68 0.10 0.46 0.46 0.04 0.35 0.63 0.32 4.65 0.67 2.61 3.39 0.73 4.37 0.82 0.62 0.22 3.45 0.64 2.56 3.40 0.64 2.50 0.82 0.62 2.46 1.33 1.65 3.21 2.53 1.60 4.19 2.28 1.28 8.50-2404.49 -2401.60 -2455.96 -2362.15 -2337.00 -2373.79 0.95 1.69 865.22 0.26 16.91 298.92 191.80 0.36 0.73 0.58 0.69 0.11 0.47 0.46 0.05 0.36 0.63 0.32 4.68 0.67 2.64 3.41 0.73 4.39 0.82 0.63 0.22 3.47 0.64 2.58 3.42 0.64 2.53 0.82 0.62 2.49 1.35 1.68 3.22 2.55 1.63 4.20 2.29 1.29 10.96-2398.40 -2397.53 -2453.52 -2360.41 -2335.90 -2373.32 1.03 2.21 101.51 0.27 17.40 304.32 197.75 0.37 0.74 0.59 0.69 0.11 0.47 0.46 0.05 0.36 0.64 0.32 4.71 0.67 2.67 3.43 0.73 4.41 0.82 0.63 0.22 3.50 0.64 2.61 3.44 0.64 2.55 0.82 0.63 2.51 1.36 1.72 3.23 2.58 1.65 4.21 2.30 1.30 11.82-2408.30 -2404.31 -2451.08 -2358.67 -2334.79 -2372.86 1.12 2.73 136.29 0.27 17.89 309.73 203.70 0.38 0.76 0.59 0.70 0.12 0.47 0.47 0.05 0.36 0.64 0.32 4.74 0.67 2.70 3.45 0.73 4.42 0.82 0.63 0.22 3.52 0.64 2.64 3.46 0.64 2.57 0.82 0.63 2.53 1.38 1.75 3.24 2.60 1.68 4.22 2.32 1.31 8.06-2402.21 -2400.24 -2462.69 -2356.93 -2333.68 -2372.39 1.21 3.25 171.08 0.28 18.39 315.14 209.64 0.39 0.77 0.59 0.71 0.12 0.47 0.47 0.05 0.36 0.65 0.32 4.77 0.67 2.73 3.47 0.73 4.44 0.82 0.63 0.22 3.54 0.64 2.66 3.48 0.65 2.60 0.82 0.63 2.55 1.39 1.78 3.25 2.63 1.71 4.22 2.33 1.33 10.77-2403.35 -2401.82 -2456.69 -2358.54 -2331.11 -2375.00 1.25 5.00 500.00 0.23 13.72 270.00 182.88 0.35 0.75 0.60 0.75 0.15 0.50 0.50 0.08 0.40 0.75 0.34 5.50 0.70 3.50 4.00 0.78 5.00 0.70 0.70 0.25 4.50 0.73 4.00 4.50 0.75 4.00 0.85 0.70 4.00 3.00 8.00 4.00 3.50 4.00 5.00 3.50 2.50 9.16

Obj. FnExperiment

242526

151617181920212223

6789

1011121314

12345

ResultsThe three studied experimental design techniques are applied, the scoping runs for each method are built, and the corresponding objective functions are calculated. ANN models are built and then minimized using genetic algorithm. The optimization algorithm, genetic algorithm, is run five times to take more representative performance indicators. The average value of the performance indicators out of the five runs is used for our comparison.

Table 5 represents the values of the error indicator obtained for the three studied experimental design techniques and for comparison purposes, the calculated results are plotted in Fig. 9. As shown in Fig. 9, Sobol and Halton sequences have significantly lower value of the average error than Latin hypercube method. Given that the same algorithm was used for proxy model generation and the same algorithm was used for optimization, the experimental design algorithm is responsible for the superior results of Sobol and Halton techniques over Latin hypercube.

SPE-Number-MS 11

Table 5: Performance indicators calculations

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35 X36 X37 X38 X39 X40 X41 X42 X43 X44 X45 X46 X47 X48 X49 X500.10 0.08 0.30 0.24 0.15 0.01 49.20 794.99 285.43 37.67 15.75 1.94 37.89 75.44 25.04 27.88 15.89 32.85 32.56 6.65 2.89 3.11 13.46 3.31 49.16 4.13 4.02 7.71 8.67 0.11 7.75 1.48 30.46 24.41 24.40 69.01 72.62 14.63 10.45 1.36 0.55 48.67 71.56 14.26 23.89 51.37 0.69 9.55 9.15 72.17 41.900.41 0.14 0.03 0.09 0.17 0.08 46.14 406.87 260.97 9.23 55.13 41.34 9.27 57.41 3.67 21.99 21.72 2.89 15.13 29.62 30.33 9.94 6.90 10.68 37.26 4.27 33.17 44.03 2.56 4.57 5.28 5.93 20.84 6.15 24.14 73.64 96.83 18.21 17.17 6.15 5.60 0.59 18.81 42.55 21.61 15.20 69.83 21.10 19.23 33.14 33.760.16 0.13 0.24 0.07 0.12 0.16 39.62 897.62 325.04 7.30 23.30 16.56 9.08 74.19 27.14 28.45 9.55 14.47 31.97 5.94 87.56 17.41 21.05 11.24 50.48 6.53 29.53 5.10 8.42 10.42 10.78 10.24 34.57 23.01 24.92 58.90 41.45 8.77 43.51 2.92 9.82 4.54 50.85 7.38 23.61 39.52 74.72 49.48 9.43 40.09 46.550.22 0.19 0.27 0.03 0.23 0.06 1.47 867.63 294.86 12.27 33.84 20.16 23.71 58.13 23.97 9.99 20.08 7.85 4.71 14.10 40.94 29.99 3.09 2.99 19.13 3.13 34.46 10.96 6.25 25.50 9.70 5.03 8.21 23.24 2.67 89.00 30.88 20.84 6.72 1.41 1.38 14.10 39.81 35.62 23.90 39.97 61.20 9.75 9.36 20.29 40.470.33 0.14 0.16 0.01 0.28 0.03 72.30 446.47 260.80 31.02 4.24 5.18 45.03 9.01 12.81 1.67 26.18 35.75 25.07 6.79 14.96 29.02 15.02 2.96 18.28 6.46 27.88 48.07 3.43 32.61 1.00 3.23 20.83 21.88 16.17 86.38 55.95 0.62 35.81 2.54 16.52 4.91 35.98 79.83 7.44 9.59 82.86 36.74 26.20 7.30 34.67

Average 0.243 0.137 0.200 0.087 0.188 0.067 41.745 682.714 ####### 19.497 26.452 17.038 24.997 54.835 18.525 17.996 18.685 18.760 21.889 12.621 35.335 17.894 11.903 6.235 34.861 4.902 25.810 23.174 5.866 14.641 6.900 5.180 22.983 19.738 18.461 75.387 59.544 12.614 22.732 2.876 6.775 14.560 43.404 35.926 20.090 31.131 57.861 25.325 14.675 34.600 39.470

0.29 0.25 0.24 0.08 0.23 0.02 87.07 336.18 18.53 10.00 20.72 37.32 20.74 72.98 27.90 15.76 3.83 24.32 10.46 3.57 16.41 8.51 16.92 3.26 31.47 5.78 30.17 2.03 6.01 16.15 4.07 13.05 5.89 3.02 2.17 0.40 89.74 4.72 64.28 2.49 2.04 1.44 8.12 4.31 5.26 28.69 82.90 23.99 1.90 6.53 23.640.15 0.08 0.28 0.04 0.15 0.00 42.08 614.72 42.89 1.77 1.59 22.24 33.46 14.81 13.38 3.21 9.42 11.28 8.29 5.68 6.26 1.14 20.32 7.75 36.03 1.32 62.75 48.06 12.24 15.88 3.42 2.19 0.32 23.69 21.46 44.88 43.32 31.51 15.92 0.28 4.48 71.98 48.57 23.95 22.48 41.09 22.00 0.52 17.53 38.02 30.300.26 0.28 0.26 0.05 0.11 0.16 45.67 494.47 10.63 19.63 46.38 17.57 36.83 23.79 9.75 18.31 2.41 38.24 9.82 18.63 88.53 10.73 20.19 13.34 70.15 5.48 33.29 41.25 1.81 27.78 11.97 6.62 3.69 39.59 16.37 17.81 19.39 0.24 54.15 0.18 7.31 2.72 8.31 24.88 18.58 14.35 21.59 46.76 15.27 67.40 30.060.26 0.28 0.26 0.05 0.11 0.16 45.67 494.47 10.63 19.63 46.38 17.57 36.83 23.79 9.75 18.31 2.41 38.24 9.82 18.63 88.53 10.73 20.19 13.34 70.15 5.48 33.29 41.25 1.81 27.78 11.97 6.62 3.69 39.59 16.37 17.81 19.39 0.24 54.15 0.18 7.31 2.72 8.31 24.88 18.58 14.35 21.59 46.76 15.27 67.40 30.060.20 0.07 0.31 0.30 0.09 0.06 48.55 365.77 58.20 11.92 52.87 2.91 91.62 15.42 2.74 13.25 12.73 52.23 4.44 9.15 57.53 10.79 10.31 5.71 73.95 2.78 27.93 47.34 1.73 31.45 0.88 10.12 2.97 9.98 2.32 15.33 75.47 21.72 51.91 2.44 21.03 24.59 30.10 17.69 15.07 26.74 31.71 32.10 14.00 99.46 30.36

Average 0.23 0.19 0.27 0.10 0.14 0.08 53.81 461.12 28.18 12.59 33.59 19.52 43.90 30.16 12.71 13.77 6.16 32.86 8.57 11.13 51.45 8.38 17.59 8.68 56.35 4.17 37.49 35.99 4.72 23.81 6.46 7.72 3.31 23.18 11.74 19.25 49.46 11.69 48.08 1.12 8.43 20.69 20.68 19.14 16.00 25.04 35.96 30.03 12.80 55.76 28.88

0.01 0.19 0.22 0.25 0.00 0.00 40.40 751.53 29.23 13.48 27.00 13.03 67.54 69.04 32.26 14.72 13.53 55.06 24.04 21.29 77.54 29.02 36.14 13.52 20.36 7.11 10.69 39.31 0.44 7.49 9.98 1.66 6.69 17.44 4.74 67.62 81.04 20.85 62.95 7.05 24.69 33.24 19.06 53.82 13.44 24.31 26.99 33.85 14.62 10.33 38.980.05 0.27 0.14 0.05 0.32 0.09 10.54 796.72 16.23 23.61 19.35 34.95 59.37 4.44 30.46 26.22 13.32 85.20 29.14 17.46 34.77 7.84 3.34 4.92 32.00 2.11 53.34 47.43 0.25 28.18 10.57 11.94 8.67 14.66 13.79 52.41 94.67 18.21 1.02 8.83 1.93 10.98 16.01 69.24 23.84 17.10 0.24 44.37 16.73 15.14 36.650.03 0.07 0.44 0.29 0.11 0.04 58.41 272.33 40.02 13.01 9.16 4.19 40.61 55.54 43.55 25.39 9.11 3.71 13.25 24.31 21.88 2.13 24.26 3.53 70.61 4.91 39.63 35.34 10.67 0.77 8.19 2.83 18.37 4.39 23.49 55.58 22.70 9.04 62.48 4.03 10.77 81.79 65.04 38.86 5.86 23.58 77.99 13.41 6.20 43.42 28.110.08 0.28 0.02 0.08 0.07 0.01 30.90 294.17 67.53 17.51 20.90 25.96 29.36 41.84 0.72 17.62 18.88 87.85 1.03 29.96 33.52 14.69 0.44 8.52 2.74 5.77 26.74 41.56 11.66 31.50 1.78 3.94 3.82 7.96 6.22 75.09 18.12 26.46 64.64 3.40 6.09 28.96 18.57 81.29 4.11 22.48 30.39 49.38 41.85 42.73 27.980.31 0.38 0.04 0.03 0.29 0.15 33.00 401.18 32.56 16.77 37.13 49.09 13.91 64.69 42.59 13.07 10.94 7.45 23.61 9.69 17.65 17.78 12.83 1.55 2.16 6.90 56.31 19.07 3.81 29.51 1.90 4.04 1.60 24.69 3.91 48.03 6.40 17.22 51.23 4.12 18.92 15.14 30.33 3.07 20.79 31.78 74.64 32.91 11.34 44.88 27.43

Average 0.10 0.24 0.17 0.14 0.16 0.06 34.65 503.19 37.11 16.87 22.71 25.44 42.16 47.11 29.92 19.41 13.16 47.85 18.22 20.54 37.07 14.29 15.40 6.41 25.57 5.36 37.34 36.54 5.37 19.49 6.48 4.88 7.83 13.83 10.43 59.75 44.59 18.36 48.46 5.48 12.48 34.02 29.80 49.26 13.61 23.85 42.05 34.78 18.15 31.30 31.83

Latin Hybercube

Sobol Sequence

Halton Sequence

Average Error Indicator, %

Average Error Indicator, %Experimental DesignTechnique

Fig. 9: Average performance indicator comparison for the studied experimental design techniques

ConclusionsBased on the results obtained from this work of comparing Sobol and Halton sequences experimental design techniques with more widely used experimental design techniques in assisted history matching of a reservoir simulation model and the results obtained from the authors’ previous work, Shams et al. (2017), we can validate the following conclusions:

Both Sobol and Halton sequences experimental design techniques are remarkably superior over the most widely used sampling technique, Latin hypercube.

Sobol and Halton sequences sampling technique give solutions closer to the exact solution, and consequently improve the assisted history matching process.

12 SPE-KSA-8665-MS

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