research article examination of experimental designs and...

Research ArticleExamination of Experimental Designs and Response SurfaceMethods for Uncertainty Analysis of Production Forecast ANiger Delta Case Study

Akeem O Arinkoola1 and David O Ogbe12

1Department of Petroleum Engineering African University of Science and Technology (AUST) Km 10 Airport Road GaladimawaAbuja Nigeria2No 1 Odi Street Old GRA Port Harcourt Rivers State Nigeria

Correspondence should be addressed to Akeem O Arinkoola moranroolaakeemyahoocom

Received 18 November 2014 Accepted 22 February 2015

Academic Editor Mikhail Panfilov

Copyright copy 2015 A O Arinkoola and D O Ogbe This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The purpose of this paper is to examine various DoE methods for uncertainty quantification of production forecast duringreservoir management Considering all uncertainties for analysis can be time consuming and expensive Uncertainty screeningusing experimental design methods helps reducing number of parameters to manageable sizes However adoption of variousmethods is more often based on experimenter discretions or company practices This is mostly done with no or little attentionbeen paid to the risks associated with decisions that emanated from that exercise The consequence is the underperformance ofthe project when compared with the actual value of the project This study presents the analysis of the three families of designsused for screening and four DoE methods used for response surface modeling during uncertainty analysis The screening methods(sensitivity by one factor at-a-time fractional experiment and Plackett-Burman design) were critically examined and analyzedusing numerical flow simulation The modeling methods (Box-Behnken central composite D-optima and full factorial) wereprogrammed and analyzed for capabilities to reproduce actual forecast figuresThe best method was selected for the case study andrecommendations were made as to the best practice in selecting various DoE methods for similar applications

1 Introduction

The increasing necessity for systematic and consistency ofmethodologies for analyzing uncertainties and their asso-ciated effects on investment decisions has become veryimportant in management decisions This is one of the majorreasons why the use of experimental design has been on theincrease in the exploration and production (EampP) industryTo be able to harness the benefits associated with design ofexperiment (DoE) usersmust apply it wisely failure of whichcan result in waste of time and costly decisions Importantareas of application of DoE in petroleum engineering includeuncertainty screening and quantification With DoE onecan significantly reduce the number of simulations requiredto assess uncertainties since considering large number ofsimulations is not feasible due to limited available time

and resources Previous screening methodologies [1] haveconsidered static sensitivity analysis Though still applicablefor some studies the dynamic behaviour of the sensitivity isimportant to observe because different uncertainty attributesmay influence reservoir performance at different time

Recent research and applied reservoir characterizationand modelling projects have focused on the application ofexperimental design (ED) and response surface techniques[2ndash4] Successes recorded from the use of ED depend largelyon number of factors These include available time num-ber of factors the study objectives and amount of detailsrequired from integrated reservoir simulation studies Thereare several designs Morris [5] classified ED into fractionalfactorial design (FRFD) optimal design and Latin hypercubedesign (LHD)The first two classes are extensively used in thearea of physical experiments for model development [6 7]

Hindawi Publishing CorporationJournal of Petroleum EngineeringVolume 2015 Article ID 714541 16 pageshttpdxdoiorg1011552015714541

2 Journal of Petroleum Engineering

LHD is an efficient design that allows representation of entireparameter range in the design matrix Its construction isbased on the idea of stratified sampling which requires asearch algorithms and prespecified optimality criteria [6]

The underlining assumption using 2-level factorialdesigns is that the 3-factor interactions higher order factorsare negligible Consequently most of the experiments areomitted from the analysis and therefore seldom used forconstruction of response surfaces For illustration if weconsider 7 parameters system that require a 27 = 128-run fulldesign Among the 127 degrees of freedom in the design only7 are used to estimate the linear terms and 21 to estimate the2-factor interactions The remaining 119 degrees of freedomare associated with three-factor interactions higher orderfactors that are supposed to be negligible In practice if thenumber of factors ranges from 2 to 15 fractional factorialcan be used to screen the significant few by estimating themain effects and interactions There are various possiblefraction designs 64- 32- or 16-run designs The 16-rundesign usually is preferred since this is the cheapest designwith resolution IV that is available for uncertainties higherthan eight

Two-level Plackett-Burman (PB) designs have been usedin the context of classical statistical experiments for screeningpurposes However PB designs cannot be used when highernumber of levels is required or where simulator proxyrsquosdevelopment remains the priority of the study One variableat a time is a screening process involving varying a parametervalues between its minimum (minus1) and maximum (+1) rangeswithin the parameter space The degree of the displacementfrom the base case of resultant response indicates the extent ofinfluence the parameter has on the measured response Onevariable at a time method is a static sensitivity common inclassical laboratory experiments However its application forscreening large number of uncertainties has been reported[8]

Three-level experimental design methods are high reso-lution algorithms for response surface construction Centralcomposite design (CCD) consists of a factorial design withthe corners at +1 of the cube augmented by additionalldquostarrdquo and ldquocentrerdquo points which allow the estimation ofthe second-order polynomial equation [9] Like CCDs Box-Behnkendesigns are a family of three-level designs and can beconstructed by the combination of factorial and incompleteblock designs [7] The application of 3-level full factorial islimited only to four factors or lower due to the accompaniedlarge number of runs for large number of uncertaintiesHowever 2-level factorial designs (fractional designs) havealso been used where 3-level full factorial is infeasible forresponse surface construction [10]

There are various response modeling techniquesdescribed in the literature [7 11] The creation of proxy-models of a high quality is related to the selection ofappropriate experimental design algorithms the quality ofthe input dataset and the degree of linearity and nonlinearityof input-output relationship Yeten et al [12] studied thecapability of different experimental designs and proxymodels to predict uncertainties A good result was reportedin application of Kriging and Splines models with Latin

hypercube design However underperformance of modelsdeveloped using traditional experimental designs wasobserved due to under sampling of uncertainty space andhigh degree of nonlinearity of input-output relationship

Fetel and Caumon [13] demonstrated how flow simula-tion results can be combined with secondary information toovercome sampling problem for modeling complex responsesurfaces A more accurate response surface at affordablecost was generated when compared with the conventionaltechniques Recent studies have shown that the problem ofnonlinearity is addressed using ANN [14] and fuzzy inferencesystem [15] that employs hybrid-learning rules for trainingprocess

Whenever experimental design is used RSM are usu-ally constructed with regression interpolation and neuralnetwork [16 17] methods The methodology integrated 3-level designs with some screening linear designs In reservoirsimulation a response surface model is essentially an equa-tion derived from the multiple regressions of all the mainattributes (reservoir parameters) that impact the reservoirresponse This provides a good approximation to simulatorand also provides simplified solution to even very complexproblems that required huge computation time [18] Theresulting proxy model is a function of the number of valuesfor each uncertainty and is readily amenable to sampling byMonte Carlo simulation for uncertainty quantification

This study critically examined three common families ofexperimental designs used for screening during uncertaintyanalysis in simulation for reservoir management Theseinclude (a) sensitivity by one factor at-a-time (b) fractionalexperiment and (c) Placket-Burman design The selectionof ldquoheavy-hittersrdquo was based on the dynamic sensitivity withresponses measured at the end-of-simulation However thepotential of missing out some parameters that are timedependent as pointed out by Amudo et al 2008 [4] wasminimized by taking measurement of the response after 15and 30 years of simulation With this measure uncertaintypropagation was adequately captured and all uncertaintieswere included in the analysis The study developed possibleresponse surface models that are consequence of the variousscreening methodsThemodels were validated and subjectedto statistical error and Monte Carlo analysis

2 Methodology

21 Description of Uncertainty There are total of 10 majoruncertainties in this study Table 1 shows the descriptions thekeywords used and the corresponding ranges of uncertain-ties as multipliers on the base case value

22 Experimentation Experiments were performed usinghistory matched model The response is the cumulativeoil production (FOPT) after 15 and 30 years of forecastSimulation runs vary according to design methods and atthe end of simulation responses are available for statisticalanalysis Figure 1 shows a typical cumulative productioncurve at the end of history or beginning of productionforecast The profile shows the existence of uncertainty in the

Journal of Petroleum Engineering 3

Table 1 Experimental range in terms of multipliers on the base case uncertain parameters

SN Parameters Keywords Minimum value Base case Maximum value1 Oil viscosity OVISC 090 1 1102 Horizontal permeability PERMX 057 1 1293 Vertical permeability PERMZ 050 1 6004 Porosity PORO 090 1 1105 Critical gas saturation SGCR 050 1 1506 Critical water saturation SWCR 053 1 1077 Fault transmissibility multiplier MULTFLT 050 1 2008 Water relative permeability KRW(SORW) 036 1 1259 Initial water saturation SWI 065 1 09010 Aquifer pore volume AQUIPV 085 1 135

5000 7500 10000 12500 15000 17500 20000

FOPT

(STB

)

Days

History match Forecast period

1E + 07

2E + 07

2E + 07

3E + 07

3E + 07

4E + 07

4E + 07

Figure 1 A typical production profile showing the end of historymatch (calibrated model) and beginning of production forecast(start date for all experiments)

response Only runs that preserved original history matchwere retained in the design matrix

23 Uncertainty Screening The three screening designsexamined in this study include (a) sensitivity by one factorat-a-time (b) fractional experiment and (c) Placket-BurmandesignUsing the parameters inTable 1 the design pointswithcorresponding responses for different methods are shown inTables 2 3 and 4

The ldquo1rdquo ldquominus1rdquo and ldquo0rdquo represent absolute high low andbase case values of the parameters The contribution of eachuncertainty factor was estimated following the significant testfor the regression used in the analysis of variance (ANOVA)for all the methods The two hypotheses used for the test areas follows

(1) The null hypothesis all treatments are of equal effects

119867

0 1198871= 119887

2= 119887

119896= 0 (1)

(2) The alternative hypothesis some treatment is ofunequal effects

119867

1 119887119895= 0 for at least one 119895 (2)

To reject the null hypothesis 1198670at least one of the variables

explains significantly the variability observed on the responseso that the model is valid

The effects of the different attributes on the response forTable 4 were estimated using [19]

Attributeeffect =119883max minus 119883minsum

1003816

1003816

1003816

1003816

119883max minus 119883min1003816

1003816

1003816

1003816

(3)

where 119883max and 119883min are maximum and minimumresponses respectively

24 Development of Response Surfaces Figures 2 3 and4 show Pareto charts associated with different screeningmethods highlighting the impact of the various parameterson the reserves over the forecast periods The vertical blacklines correspond to a 95 confidence level implying that anyparameter to the right is significant (ldquoheavy-hitterrdquo) with 95confidence

Figure 2 shows the six ldquoheavy-hittersrdquo identified forfractional experiment at the end of 15-year forecast theOVISC SWI PERMX PORO KRW and PERMZ The fourother parameters that is the SGCR theAQUIPVMULTFLTand SWCR are insignificant on reserve forecast within theseperiods However at the end of 30-year simulation KRWeffect was insignificant and this reduced the number ofldquoheavy-hittersrdquo to five

Figure 3 shows the five ldquoheavy-hittersrdquo identified for PBexperiment at the end of 15-year forecast the OVISC SWIPERMX PORO and PERMZ However at the end of 30-yearsimulation PERMZ became insignificant on the model Thefour identified ldquoheavy-hittersrdquo includeOVISC SWI PERMXand PORO

Figure 4 shows the ldquoheavy-hittersrdquo identified for varyingone parameter at-a-time at the end of 15- and 30-yearforecast respectively In both cases OVISC PERMX andSWI have the most influence on the response One importantobservation is the time-dependent nature of the aquiferproperties (AQUIPV)

To determine whether there is a linear relationshipbetween the response and various ldquoheavy-hittersrdquo test for sig-nificance of regressionwas performed byANOVAThis studyutilized the computed Fisher variable (119865) and its associatedprobability (prob gt 119865) to assess the model significance aswell as to select regressors with high impact on the model byassuming 95 confidence level or 5 significant limit (ie120572 = 005)


Table2DoE

matrix

forfractionalfactoria

lmetho

d

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

Respon

seFOPT

(STB

)15

years

30years

1minus1

11

1minus1

1minus1

minus1

minus1

minus1

2927126833277150

21

minus1

11

minus1

minus1

1minus1

minus1

minus1

2561077628803106

3minus1

11

minus1

minus1

minus1

11

1minus1

3055836634219088

4minus1

minus1

11

1minus1

minus1

11

13063223036143012

51

1minus1

1minus1

minus1

minus1

1minus1

12738475831335538

6minus1

1minus1

11

minus1

1minus1

1minus1

3032718235035264

7minus1

1minus1

minus1

11

minus1

1minus1

minus1

2811745031376578

8minus1

minus1

minus1

1minus1

11

1minus1

12782702832052148

91

11

minus1

1minus1

minus1

minus1

minus1

12648546830122768

10

minus1

minus1

1minus1

11

1minus1

minus1

12707957831057656

11

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

11

2734039031168096

12

1minus1

minus1

minus1

1minus1

11

minus1

minus1

2530320827984070

13

1minus1

1minus1

minus1

1minus1

11

minus1

2682741029853822

14

11

minus1

minus1

minus1

11

minus1

11

2734492031090104

15

1minus1

minus1

11

1minus1

minus1

1minus1

2652291630021116

16

11

11

11

11

11

3013041635331656


Table3PB

desig

ntablefor

10parameters

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

FOPT

(15y

rs)

FOPT

(30y

rs)

1minus1

minus1

minus1

11

1minus1

11

minus1

30848400

33479500

21

minus1

11

minus1

1minus1

minus1

minus1

127154800

29412400

31

11

minus1

11

minus1

1minus1

minus1

28005900

29769600

41

1minus1

11

minus1

1minus1

minus1

minus1

27889000

30076700

5minus1

minus1

11

1minus1

11

minus1

130629500

33815700

6minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

26898400

28727400

71

minus1

1minus1

minus1

minus1

11

1minus1

28034500

29923100

81

minus1

minus1

minus1

11

1minus1

11

27275300

29486200

9minus1

11

minus1

1minus1

minus1

minus1

11

32087300

35060000

10

minus1

1minus1

minus1

minus1

11

1minus1

130096500

32551700

11

minus1

11

1minus1

11

minus1

1minus1

33201400

36053900

12

11

minus1

1minus1

minus1

minus1

11

130582600

33411100


Table 4 One variable at a time design table for 10 parameters

Runs OVISC PERMX PERMZ PORO SGCR SWCR MULTFLT KRW SWI AQUIPV FOPT (15 yrs) FOPT (30 yrs)1 1 0 0 0 0 0 0 0 0 0 29761616 352456722 minus1 0 0 0 0 0 0 0 0 0 29116404 344894523 0 1 0 0 0 0 0 0 0 0 29343824 344957684 0 minus1 0 0 0 0 0 0 0 0 29058182 342264525 0 0 1 0 0 0 0 0 0 0 28566772 333224166 0 0 minus1 0 0 0 0 0 0 0 29484880 347396167 0 0 0 1 0 0 0 0 0 0 29502398 351384768 0 0 0 minus1 0 0 0 0 0 0 29016448 342281369 0 0 0 0 1 0 0 0 0 0 28963490 3410983610 0 0 0 0 minus1 0 0 0 0 0 28046878 3273712211 0 0 0 0 0 1 0 0 0 0 28873100 3353829612 0 0 0 0 0 minus1 0 0 0 0 28597504 3366192413 0 0 0 0 0 0 minus1 0 0 0 28902034 3403426814 0 0 0 0 0 0 1 0 0 0 30595368 3619320015 0 0 0 0 0 0 0 minus1 0 0 27848692 3256835016 0 0 0 0 0 0 0 1 0 0 28692348 3376908417 0 0 0 0 0 0 0 0 minus1 0 28460282 3308352818 0 0 0 0 0 0 0 0 1 0 28978374 3397574019 0 0 0 0 0 0 0 0 0 minus1 28928676 3399166420 0 0 0 0 0 0 0 0 0 1 26487010 30639428

Significance limit

0 5 10 15 20 25 30 35AE

FAHAGADAC

G-MULTFLTK-AQUPV

E-SGCRC-PERMZ

H-KRWD-PORO

B-PERMXJ-ISW

A-OVISC

Contribution ()

Para

met

ers

(a)

0 5 10 15 20 25 30 35

AEF

AHACAGAD

G-MULTFLTE-SGCRH-KRW

K-AQUPVC-PERMZB-PERMX

D-POROJ-ISW

A-OVISC

Contribution ()

Para

met

ers

Significance limit

(b)

Figure 2 Pareto charts from fractional experiment showing key parameters impacting reserves after (a) 15-year and (b) 30-year forecasts

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTK-AQUPV

H-KRWC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution

Para

met

ers

Significant

(a)

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTH-KRW

K-AQUPVC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution ()

Para

met

ers

Significant

(b)

Figure 3 Pareto charts from Placket-Burman experiment showing key parameters impacting reserves after (a) 15-year and (b) 30-yearforecasts


0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)

Figure 4 Pareto chart from one parameter at-a-time experiment showing key parameters impacting reserves after (a) 15-year forecast and(b) 30-year forecast

Table 5 Analysis of variance for Box-Behnken model associated with fractional factorial screening design

Source Sum of squares DF Mean square 119865-value Prob gt 119865Model 166119864 + 14 9 185119864 + 13 103125 lt00001 SignificantA-PORO 158119864 + 13 1 158119864 + 13 88142 lt00001B-PERMX 181119864 + 13 1 181119864 + 13 101313 lt00001C-PERMZ 753119864 + 12 1 753119864 + 12 42069 lt00001D-ISW 178119864 + 13 1 178119864 + 13 99428 lt00001E-OVISC 323119864 + 13 1 323119864 + 13 180644 lt00001B2

187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10

Pure error 0 5 0Cor total 167119864 + 14 45

Table 5 is a typical ANOVA table for Box-Behnkenmethod associated with fractional factorial screening designThemodel 119865-value of 103125 implies the model is significantThere is only a 001 chance that a ldquomodel 119865-valuerdquo this largecould occur due to noise Values of ldquoProb gt 119865rdquo less than00500 indicate models terms are significant In this case A(PORO) B (PERMX) C (PERMZ) D (SWI) E (OVISC) B2C2 D2 and E2 are all significant model terms

In general 10 different response surface correlationswere developed The regression equations were based onthe outcome from the three screening methods earlier dis-cussed Four methods were considered for response surfacemodelling Box-Behnken design (BBD) central compositedesign (CCD) full factorial design (FFD) and D-optimaldesign (DOD) Thus model equations (4) (5) (6) and (7)are associated with fractional factorial method (FRFDRSMs)model equations (8) (9) (10) and (11) are associated withPlacket-Burman method (PBDRSMs) and the 2 feasibleresponse model equations (12) and (13) are associated withscreening using one variable at-a-time (OVAATRSMs)

Tables 6 7 and 8 show the constants in all the equations

FOPTBox-Behnken (MMSTB)

= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)

FOPTCentral Composite (MMSTB)

= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)

FOPTD-Optimal (MMSTB)

= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)


FOPTFull Factorial (MMSTB)

= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887

9PORO lowast SWC + 119887

10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation

Figures 5(a) 5(b) and 5(c) are parity plots for the variousprediction methods corresponding to different screeningmethod These graphs are plots of the predicted productionforecast as a function of the experimental reserves values If

Table 6 Constants in (4) (5) (6) and (7)

119886

119894Box-Behnken Central composite D-optimal Full factorial119886

0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886

5minus13830 minus1280 minus6358 minus104

119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171

Table 8 Constants in (12) and (13)

119862

119894Box-Behnken method D-optimal method

119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000

the prediction methods were a perfect fit of the experimentaldata then all of the points would lie on the 119909 = 119910 line

Parity plots do not reveal much information Howeverthe plots for the Box-Behnkenmethod generally demonstratethat the method predict the actual value more accuratelyOn these plots the vast majority of the points are along the119909 = 119910 line In addition the plots reveal that regardlessof the screening method CCD exhibits the least accurateprediction

The validation using cross plots only assess model effi-ciencies within the experimental range of parameters In


Pred

icte

d va

lue (

MM

STB)

Actual value (MMSTB)

3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07

Box-BehnkenFull factorial

D-optimalCentral composite

(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07

Box-Behnken (R2= 0982)

D-optimal (R2= 0972)


(c)

Figure 5 Comparison of the actual and predicted reserves by (a) FRFRSMs (b) PBRSMs and (c) OVAATRSMs

order to determine model predictability elsewhere this studyperformed a ldquoblind testrdquo using parameter values outside therange already defined in Table 1 The simulation and predic-tion were made only on OVISC because it is controllable

Figures 6(a) 6(b) and 6(c) show respectively the com-parison of RSMs from FRFD PBD and OVAAT with theactual experimentalsimulated valuesThe simulated produc-tion upon degradation of OVISC by 30 50 and 70 isshown in Figure 6(d) It was noticed that prediction usingfull factorial method was excellent with maximum deviationof 079MMSTB followed by central composite method withmaximum deviation of 39MMSTB Box-Behnken generallyoverpredicted the actual reserves volume with deviationapproximately 20MMSTB

4 Statistical Error Analysis

The performance indices used are summarized in Table 9Each of the response surfaces developed was used to estimatethe production forecast for each of the data sets Tables 10 11

Table 9 Performance indices for model evaluation

Name of measure Formula

Absolute deviation AD = 1119873

119873

sum

119894=1

(Pred minus Exp)

Average absolutedeviation AAD = 1

119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816

Root mean squareerror RMSE = radic 1

119873

119873

sum

119894=1

(Actual minus Predicted)2

Average absolutepercentage relativeError

AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]

Maximum error 119864max = Max 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100

Standard deviation SD = 1119873

119889minus 1

lowast

119873

sum

119894=1

119864

119894

2

and 12 show the result from the error analysis In additionto the estimated errors the coefficients of correlations are


0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)

Oil viscosity degraded ()

Box-Behnken Central compositeD-optimal Full factorialSimulation

(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)

30 50 70Oil viscosity degraded ()

Box-BehnkenD-optimalSimulation

(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days

70 degraded50 degraded30 degraded

3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)

Figure 6 Comparison of the predictions using (a) FRFRSMs (b) PBRSMs and (c) OVAATRSMs (d) the simulated reserves upon reductionof OVISC by 30 50 and 70

Table 10 Summary of the statistical analysis of RSMs from fractional factorial screening

Performance index Box-Behnken Full factoriallowast D-optimal Central compositeRMSE 1185877 1814203 1816590 3020953

AAD 845652 1413333 1533333 2276923

AD minus6522 minus6667 7407 minus15385

119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470

lowast2- level full factorial was used


Table 11 Summary of the statistical analysis of RSMs from Placket-Burman screening

Performance index Box-Behnken Full factorial D-optimal Central compositeRMSE 1253281 1242645 2182697 6085815

AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250

Table 12 Summary of the statistical analysis of RSMs fromOVAATscreening

Performance index D-optimal Box-BehnkenRMSE 3458238 2823816AAD 2811765 2152459AD 5882 minus8197

119864max 21 23SD 11 07AAPRE 09 06119877-square () 972 982Adj 119877-square () 963 977Pred 119877-square () 951 967Exp runs 370 620

also indicated Box-Behnken method exhibited the least esti-mation error with highest values of correlation coefficientsCCD on the other hand showsmaximum estimated error andleast correlation coefficients Again this was performed onpredictions within the actual parameter range of values

5 Representative Model

As shown in Table 13 all models were ranked using thefollowing criteria number of uncertainties associated withthe different screening methods result from error analysisresult from the ldquoblindrdquo test and coefficients of correlationThe decision to select ldquobestrdquo model was based on scenarioobjectives

51 Objective 1 If it is desired to develop risk curvesfrom where P90P50P10 values for the response can bedetermined alongside its representative models the analysisfavours the selection of Box-Behnken method There arehowever three possible response surfaces depending on thescreening method These are response surface equations (4)(with 5 factors 46 runs) (8) (with 4 factors 29 runs) and(13) (with 7 factors 62 runs) For practical purposes responsesurface equation (8) is desirable with fewer number of factorsand experimental runs

52 Objective 2 If it is desired to utilized the proxy equa-tion to simulate and evaluate future development strategiessuch as the needs for acquiring additional informationundertaking stimulation or any of EOR such as in situcombustion a more efficient model capable of estimatingreservoir performance within acceptable margin of error isdesired Both central composite (CCD) and full factorialmethods are adequate

However in this analysis full factorialmethod performedfar better than the CCD FFD tends to be impractical forlarge number of uncertainties We have demonstrated in thisstudy that a 2-level factorial experiment can be used forconstructing response surface of quality as comparable as thatfrom 3-level full factorial However we highly recommendedconsideration for resolution of the factorial fractions to beused

6 Monte Carlo Simulation

The Monte Carlo technique [20] was used to combine theuncertain attributes and allows generating values for modelinput variables 2000 iterations were made while assumingthe following distribution functions triangular for OVISCuniform for PORO log normal for PERMX and PERMZnormal for SWC triangular for AQUIPV and triangular forKRWThe risk curves generated were in terms of cumulativeoil production

Figure 7 shows risk curves from various RSMs for theuncertainty assessment It is clearly shown that the P10and P90 values differ according to the assumed responsesurface methods Full factorial method gives the highestP10 (384MMSTB) value and minimum P90 (28MMSTB)value CCD is characterized by lowest P10 (354MMSTB)and highest P90 (32MMSTB) Both Box-Behnken andD-optimal methods give approximately equal values ofP90 (36MMSTB) and P10 (30MMSTB) These figuresrequire additional evaluation framework for ranking differentprospects regarding different outcomes so that feasible deci-sions when comparing risky capital investment can be madeinstead of decisions based on ldquobest guessrdquo

The costs and benefits of performing 46 and 62experiments instead of the desired 29 PB experiments


Table 13 Summary table for model ranking

Ranking measures Box-Behnken Response surface methods Full factorialCentral composite D-optimal

Fractional factorial screening InfeasiblePlacket-Burman screening

OVAAT screening Infeasible InfeasibleError analysis Best Average Average BetterBlind test Fail Fail

Correlation 119877-squares

PassInfeasible not practicable due to large number of experimental runs

0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)

Central compositeFull factorial

D-optimalBox-Behnken

384MMSTB36MMSTB

354MMSTB

Figure 7 Impact of different RSMs on uncertainty assessment

were determined by constructing multiple risk curveswith equal or about the same P50 (mean) values for allmodels

Figure 8(a) shows a good agreement of risk curves ofBox-Behnken associatedwith Placket-Burman and fractionalfactorial screening designs The same applied to full factorialrisk curves shown in Figure 8(c) However the risk curveassociated with screening using the one variable at-a-timeexhibits wider variability from others and tends to be moreoptimistic which is perhaps due to different occurrenceprobability and larger number of factors involved in theregression analysis

The risk curves obtained by CCD developed fromPlacket-Burman and fractional factorial are a little at variancefrom each other due to combination that possesses dissimilaroccurrence probability

In summary both fractional screening design and PBdesign tend to give approximate result Based on the analysisthere is no significant added advantage in performing 46experiments as required by fractional factorial screeningmethod PB with fewer experimental runs is therefore desir-able

7 Forecast Distribution

It was found that 2000 equiprobable realisations iterativelybuilt in Excel were enough to stabilize the resulting forecastdistributions One critical measure used to generate eachrealisation of the model is that the deterministic reservevalue was fairly maintained at simulation base case valuethroughout the process Cumulative probability distributionfor the forecast reserves is shown in Figure 9 The impactof the major uncertain parameters was quantified It clearlyappears in Figure 10 that the oil viscosity and water sat-uration uncertainties are the most influential parametersThe other three parameters (porosity horizontal and verticalpermeability) have less importanceThe spread in productionprofiles (P10ndashP90 range) from the deterministic forecast(P50) volume as shown in Figure 11 indicates occurrence ofuncertainty in the forecast The extreme quantities to a largeextent are investment indicators

8 Summary

(i) The major objective of this study was to investi-gate the implications of various experimental designassumptions usually made while performing uncer-tainty analysis after reservoir simulation for reservoirmanagement

(ii) The three families of screening designs considered arefractional factorial Placket-Burman and one variableat-a-time

(iii) The four response surface methods considered forthe regression modeling are full factorial centralcomposite Box-Behnken and D-optimal

(iv) A total of 9 response surface models developed werevalidated and subjected to statistical error analysis

(v) The models were ranked using some criteria andbased on case objectives selection of appropriatemodel was made

(vi) The risk curves generated were used to provide infor-mation about the costs and benefits of conductingadditional experiments due to differences in thenumber of factors emanated from using differentscreening methods


0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBDFOPT_OVAAT

(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)

Figure 8 Risk curves for Case 1 (a) Box-Behnken equations (4) (8) and (13) (b) central composite equations (5) and (11) and (c) fullfactorial equations (7) and (9)

(vii) Also the risk curve was employed to identify stochas-tic models associated with the P10P50P90 realiza-tions

9 Conclusion and Recommendations

This study examined three screening designs (Placket-Burman fractional factorial and one variable at-a-time) andfour response surface methodologies (Box-Behnken centralcomposite D-optimal and full factorial) commonly used foruncertainty analysis In all screening methods years of pro-duction forecast played important role on associated numberof ldquoheavy-hittersrdquo One variable at-a-time was identified withlargest number of parameters and hence considered notideal because of the attendant large number of simulationruns Unlike Placket-Burman a low resolution fractionalfactorial in addition to main effects considered significance

of factor interactions requiredmore simulation runs but canprevent exclusion of some factors with minimal main effectbut significant interaction effect Nevertheless the analysisperformed in this study using Monte Carlo simulation showsthat there was no added advantage using fractional factorialin lieu of Placket-Burman for screening

The ldquobestrdquo model for uncertainty quantification must beselected based on the reservoir management objectives Box-Behnken method was adequate to determine P10P50P90and associated models On the other hand to evaluate futuredevelopment strategies such as EOR stimulation and theneeds for acquiring additional information full factorial andcentral composite designs aremore efficient predictors withinacceptable margin of error A full 2-level factorial or highresolution fractional factorial method was equally adequatefor the construction of response surfaces for uncertaintyquantification where 3-level full factorial was not feasible


0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40

Reserves distribution (MMSTB)

00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves

MinimumMaximumMeanStd Dev1090Values

(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)

Figure 9 Reserves distribution for (a) Box-Behnken RSM equation (8) (b) central composite and (c) full factorial response surfacesassociated with Placket-Burman screening method


029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC

Reserves correlation coefficient (Spearman rank)

minus044

minus064

minus08 minus06 minus04 minus02

Coefficient value

Figure 10 Ranking of uncertainty impact on production forecast

12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10

Full factorialCentral compositeBox-Behnken

3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

Figure 11 Stochastic model profiles corresponded withP10P50P90 for Box-Behnken central composite and fullfactorial PB associated methods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledged Schlumberger for providing thesoftware at AUST for simulation The authors also wishto acknowledge Petroleum Technology Development Fund(PTDF) for supporting this research

References

[1] D E Steagall and D J Schiozer ldquoUncertainty analysis inreservoir production forecasts during appraisal and pilot pro-duction phasesrdquo in Proceedings of the SPE Reservoir SimulationSymposium SPE 66399 Houston Tex USA February 2001

[2] N Almeida D J Schiozer E L Ligero and C MaschioldquoHistory matching using uncertainty analysisrdquo in Proceedingsof the SPE Canadian International Petroleum Conference SPE153604 Calgary Canada June 2003

[3] CH Peng and R Gupta ldquoExperimental design in deterministicmodelling assessing significant uncertaintiesrdquo in Proceedings ofthe SPE Asia Pacific Oil and Gas Conference SPE 80537 JakartaIndonesia September 2003

[4] C Amudo T Graf N R Haris R Dandecar F Ben Amor andR S May ldquoExperimental design and response surface modelsas a basis for stochastic history matchmdasha Niger delta experi-encerdquo in Proceedings of the International Petroleum TechnologyConference (IPTC rsquo08) IPTC 12665 Kuala Lumpa MalaysiaDecember 2008

[5] M D Morris ldquoThree technometrics experimental design clas-sicsrdquo Technometrics vol 42 no 1 pp 26ndash27 2000

[6] G E P Box W G Hunter and J S Hunter Statistics forExperimenters Design Innovation and Discovery John Wileyamp Sons New York NY USA 2nd edition 2005

[7] D C Montgomery Design and Analysis of ExperimentsResponse Surface Method and Designs John Wiley amp SonsHoboken NJ USA 2005

[8] F Moeinikia and N Alizadeh ldquoExperimental Design in reser-voir simulation an integrated solution for uncertainty analysisa case studyrdquo Journal of Petroleum Exploration and ProductionTechnology vol 2 no 2 pp 75ndash83 2012

[9] T TAllenMA Bernshteyn andKKabiri-Bamoradian ldquoCon-structing meta-models for computer experimentsrdquo Journal ofQuality Technology vol 35 no 3 pp 264ndash274 2003

[10] V S Aigbodion S B Hassan E T Dauda and R AMohammed ldquoThe development of mathematical model for theprediction of ageing behavior for Al-Cu-Mgbagasse particulatecompositerdquo Journal of Minerals amp Materials Characteristics ampEngineering vol 9 pp 907ndash917 2010

[11] E Manceau M Mezghani I Zabalza-Mezghani and F Rog-gero ldquoCombination of experimental design and joint modelingmethods for quantifying the risk associated with deterministicand stochastic uncertaintiesmdashan integrated test studyrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSPE 71620 pp 2537ndash2547 NewOrleans Lo USAOctober 2001

[12] B Yeten A Castellini B Guyaguler and W H Chen ldquoAcomparison study on experimental design and response surfacemethodologiesrdquo in Proceedings of the SPE Reservoir SimulationSymposium vol 93347 pp 465ndash479 Houston Tex USA Febru-ary 2005

[13] E Fetel and G Caumon ldquoReservoir flow uncertainty assess-ment using response surface constrained by secondary infor-mationrdquo Journal of Petroleum Science and Engineering vol 60no 3-4 pp 170ndash182 2008

[14] S Mohaghegh ldquoVirtual-intelligence applications in petroleumengineering part Imdashartificial neural networksrdquo Journal ofPetroleum Technology vol 52 no 9 pp 64ndash73 2000

[15] K K SalamDAraromi and S S Ikiensikimama ldquoNeuro-fuzzymodeling for the prediction of below-bubble-point viscosityrdquoPetroleum Science and Technology vol 29 no 17 pp 1741ndash17522011

[16] KMursquoazu I AMohammed-Dabo and SMWaziri ldquoDevelop-ment of mathematical model for the prediction of essential oilextraction from Eucalyptus citriodora leavesrdquo Journal of Basicand Applied Scientific Research vol 2 no 3 pp 2298ndash23062012


[17] A Friedmann D K Chawathe and D K Larue ldquoAssess-ing uncertainty in channelized reservoirs using experimentaldesignsrdquo in Proceedings of the SPE Reservoir Evaluation ampEngineering August 2003 SPE 85117

[18] B Guyagular R N Horne L Rogers and J J RosenzweigldquoOptimization of well placement in a Gulf of Mexico Water-flooding Projectrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 63221 Dallas Tex USA Octo-ber 2001

[19] J-P Dejean and G Blanc ldquoManaging uncertainties on produc-tion predictions using integrated statistical methodsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionlsquoReservoir Engineeringrsquo vol 56696 Houston Tex USAOctober1999

[20] J M Hammersley andD C HandscombMonte CarloMethodsChapman and Hall London UK 1983

International Journal of

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



LHD is an efficient design that allows representation of entireparameter range in the design matrix Its construction isbased on the idea of stratified sampling which requires asearch algorithms and prespecified optimality criteria [6]

The underlining assumption using 2-level factorialdesigns is that the 3-factor interactions higher order factorsare negligible Consequently most of the experiments areomitted from the analysis and therefore seldom used forconstruction of response surfaces For illustration if weconsider 7 parameters system that require a 27 = 128-run fulldesign Among the 127 degrees of freedom in the design only7 are used to estimate the linear terms and 21 to estimate the2-factor interactions The remaining 119 degrees of freedomare associated with three-factor interactions higher orderfactors that are supposed to be negligible In practice if thenumber of factors ranges from 2 to 15 fractional factorialcan be used to screen the significant few by estimating themain effects and interactions There are various possiblefraction designs 64- 32- or 16-run designs The 16-rundesign usually is preferred since this is the cheapest designwith resolution IV that is available for uncertainties higherthan eight

Two-level Plackett-Burman (PB) designs have been usedin the context of classical statistical experiments for screeningpurposes However PB designs cannot be used when highernumber of levels is required or where simulator proxyrsquosdevelopment remains the priority of the study One variableat a time is a screening process involving varying a parametervalues between its minimum (minus1) and maximum (+1) rangeswithin the parameter space The degree of the displacementfrom the base case of resultant response indicates the extent ofinfluence the parameter has on the measured response Onevariable at a time method is a static sensitivity common inclassical laboratory experiments However its application forscreening large number of uncertainties has been reported[8]

Three-level experimental design methods are high reso-lution algorithms for response surface construction Centralcomposite design (CCD) consists of a factorial design withthe corners at +1 of the cube augmented by additionalldquostarrdquo and ldquocentrerdquo points which allow the estimation ofthe second-order polynomial equation [9] Like CCDs Box-Behnkendesigns are a family of three-level designs and can beconstructed by the combination of factorial and incompleteblock designs [7] The application of 3-level full factorial islimited only to four factors or lower due to the accompaniedlarge number of runs for large number of uncertaintiesHowever 2-level factorial designs (fractional designs) havealso been used where 3-level full factorial is infeasible forresponse surface construction [10]

There are various response modeling techniquesdescribed in the literature [7 11] The creation of proxy-models of a high quality is related to the selection ofappropriate experimental design algorithms the quality ofthe input dataset and the degree of linearity and nonlinearityof input-output relationship Yeten et al [12] studied thecapability of different experimental designs and proxymodels to predict uncertainties A good result was reportedin application of Kriging and Splines models with Latin

hypercube design However underperformance of modelsdeveloped using traditional experimental designs wasobserved due to under sampling of uncertainty space andhigh degree of nonlinearity of input-output relationship

Fetel and Caumon [13] demonstrated how flow simula-tion results can be combined with secondary information toovercome sampling problem for modeling complex responsesurfaces A more accurate response surface at affordablecost was generated when compared with the conventionaltechniques Recent studies have shown that the problem ofnonlinearity is addressed using ANN [14] and fuzzy inferencesystem [15] that employs hybrid-learning rules for trainingprocess

Whenever experimental design is used RSM are usu-ally constructed with regression interpolation and neuralnetwork [16 17] methods The methodology integrated 3-level designs with some screening linear designs In reservoirsimulation a response surface model is essentially an equa-tion derived from the multiple regressions of all the mainattributes (reservoir parameters) that impact the reservoirresponse This provides a good approximation to simulatorand also provides simplified solution to even very complexproblems that required huge computation time [18] Theresulting proxy model is a function of the number of valuesfor each uncertainty and is readily amenable to sampling byMonte Carlo simulation for uncertainty quantification

This study critically examined three common families ofexperimental designs used for screening during uncertaintyanalysis in simulation for reservoir management Theseinclude (a) sensitivity by one factor at-a-time (b) fractionalexperiment and (c) Placket-Burman design The selectionof ldquoheavy-hittersrdquo was based on the dynamic sensitivity withresponses measured at the end-of-simulation However thepotential of missing out some parameters that are timedependent as pointed out by Amudo et al 2008 [4] wasminimized by taking measurement of the response after 15and 30 years of simulation With this measure uncertaintypropagation was adequately captured and all uncertaintieswere included in the analysis The study developed possibleresponse surface models that are consequence of the variousscreening methodsThemodels were validated and subjectedto statistical error and Monte Carlo analysis

2 Methodology

21 Description of Uncertainty There are total of 10 majoruncertainties in this study Table 1 shows the descriptions thekeywords used and the corresponding ranges of uncertain-ties as multipliers on the base case value

22 Experimentation Experiments were performed usinghistory matched model The response is the cumulativeoil production (FOPT) after 15 and 30 years of forecastSimulation runs vary according to design methods and atthe end of simulation responses are available for statisticalanalysis Figure 1 shows a typical cumulative productioncurve at the end of history or beginning of productionforecast The profile shows the existence of uncertainty in the




5000 7500 10000 12500 15000 17500 20000

FOPT

(STB

)

Days


1E + 07

2E + 07

2E + 07

3E + 07

3E + 07

4E + 07

4E + 07






119867

0 1198871= 119887

2= 119887

119896= 0 (1)


119867

1 119887119895= 0 for at least one 119895 (2)





1003816

1003816

1003816

1003816

119883max minus 119883min1003816

1003816

1003816

1003816

(3)








Table2DoE

matrix


lmetho

d

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

Respon

seFOPT

(STB

)15

years

30years

1minus1

11

1minus1

1minus1

minus1

minus1

minus1

2927126833277150

21

minus1

11

minus1

minus1

1minus1

minus1

minus1

2561077628803106

3minus1

11

minus1

minus1

minus1

11

1minus1

3055836634219088

4minus1

minus1

11

1minus1

minus1

11

13063223036143012

51

1minus1

1minus1

minus1

minus1

1minus1

12738475831335538

6minus1

1minus1

11

minus1

1minus1

1minus1

3032718235035264

7minus1

1minus1

minus1

11

minus1

1minus1

minus1

2811745031376578

8minus1

minus1

minus1

1minus1

11

1minus1

12782702832052148

91

11

minus1

1minus1

minus1

minus1

minus1

12648546830122768

10

minus1

minus1

1minus1

11

1minus1

minus1

12707957831057656

11

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

11

2734039031168096

12

1minus1

minus1

minus1

1minus1

11

minus1

minus1

2530320827984070

13

1minus1

1minus1

minus1

1minus1

11

minus1

2682741029853822

14

11

minus1

minus1

minus1

11

minus1

11

2734492031090104

15

1minus1

minus1

11

1minus1

minus1

1minus1

2652291630021116

16

11

11

11

11

11

3013041635331656


Table3PB

desig

ntablefor

10parameters

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

FOPT

(15y

rs)

FOPT

(30y

rs)

1minus1

minus1

minus1

11

1minus1

11

minus1

30848400

33479500

21

minus1

11

minus1

1minus1

minus1

minus1

127154800

29412400

31

11

minus1

11

minus1

1minus1

minus1

28005900

29769600

41

1minus1

11

minus1

1minus1

minus1

minus1

27889000

30076700

5minus1

minus1

11

1minus1

11

minus1

130629500

33815700

6minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

26898400

28727400

71

minus1

1minus1

minus1

minus1

11

1minus1

28034500

29923100

81

minus1

minus1

minus1

11

1minus1

11

27275300

29486200

9minus1

11

minus1

1minus1

minus1

minus1

11

32087300

35060000

10

minus1

1minus1

minus1

minus1

11

1minus1

130096500

32551700

11

minus1

11

1minus1

11

minus1

1minus1

33201400

36053900

12

11

minus1

1minus1

minus1

minus1

11

130582600

33411100




Significance limit

0 5 10 15 20 25 30 35AE

FAHAGADAC

G-MULTFLTK-AQUPV

E-SGCRC-PERMZ

H-KRWD-PORO

B-PERMXJ-ISW

A-OVISC

Contribution ()

Para

met

ers

(a)

0 5 10 15 20 25 30 35

AEF

AHACAGAD



D-POROJ-ISW

A-OVISC

Contribution ()

Para

met

ers

Significance limit

(b)


0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTK-AQUPV

H-KRWC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution

Para

met

ers

Significant

(a)

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTH-KRW

K-AQUPVC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution ()

Para

met

ers

Significant

(b)



0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)




187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10






= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)



= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












5000 7500 10000 12500 15000 17500 20000

FOPT

(STB

)

Days


1E + 07

2E + 07

2E + 07

3E + 07

3E + 07

4E + 07

4E + 07






119867

0 1198871= 119887

2= 119887

119896= 0 (1)


119867

1 119887119895= 0 for at least one 119895 (2)





1003816

1003816

1003816

1003816

119883max minus 119883min1003816

1003816

1003816

1003816

(3)








Table2DoE

matrix


lmetho

d

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

Respon

seFOPT

(STB

)15

years

30years

1minus1

11

1minus1

1minus1

minus1

minus1

minus1

2927126833277150

21

minus1

11

minus1

minus1

1minus1

minus1

minus1

2561077628803106

3minus1

11

minus1

minus1

minus1

11

1minus1

3055836634219088

4minus1

minus1

11

1minus1

minus1

11

13063223036143012

51

1minus1

1minus1

minus1

minus1

1minus1

12738475831335538

6minus1

1minus1

11

minus1

1minus1

1minus1

3032718235035264

7minus1

1minus1

minus1

11

minus1

1minus1

minus1

2811745031376578

8minus1

minus1

minus1

1minus1

11

1minus1

12782702832052148

91

11

minus1

1minus1

minus1

minus1

minus1

12648546830122768

10

minus1

minus1

1minus1

11

1minus1

minus1

12707957831057656

11

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

11

2734039031168096

12

1minus1

minus1

minus1

1minus1

11

minus1

minus1

2530320827984070

13

1minus1

1minus1

minus1

1minus1

11

minus1

2682741029853822

14

11

minus1

minus1

minus1

11

minus1

11

2734492031090104

15

1minus1

minus1

11

1minus1

minus1

1minus1

2652291630021116

16

11

11

11

11

11

3013041635331656


Table3PB

desig

ntablefor

10parameters

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

FOPT

(15y

rs)

FOPT

(30y

rs)

1minus1

minus1

minus1

11

1minus1

11

minus1

30848400

33479500

21

minus1

11

minus1

1minus1

minus1

minus1

127154800

29412400

31

11

minus1

11

minus1

1minus1

minus1

28005900

29769600

41

1minus1

11

minus1

1minus1

minus1

minus1

27889000

30076700

5minus1

minus1

11

1minus1

11

minus1

130629500

33815700

6minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

26898400

28727400

71

minus1

1minus1

minus1

minus1

11

1minus1

28034500

29923100

81

minus1

minus1

minus1

11

1minus1

11

27275300

29486200

9minus1

11

minus1

1minus1

minus1

minus1

11

32087300

35060000

10

minus1

1minus1

minus1

minus1

11

1minus1

130096500

32551700

11

minus1

11

1minus1

11

minus1

1minus1

33201400

36053900

12

11

minus1

1minus1

minus1

minus1

11

130582600

33411100




Significance limit

0 5 10 15 20 25 30 35AE

FAHAGADAC

G-MULTFLTK-AQUPV

E-SGCRC-PERMZ

H-KRWD-PORO

B-PERMXJ-ISW

A-OVISC

Contribution ()

Para

met

ers

(a)

0 5 10 15 20 25 30 35

AEF

AHACAGAD



D-POROJ-ISW

A-OVISC

Contribution ()

Para

met

ers

Significance limit

(b)


0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTK-AQUPV

H-KRWC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution

Para

met

ers

Significant

(a)

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTH-KRW

K-AQUPVC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution ()

Para

met

ers

Significant

(b)



0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)




187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10






= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)



= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table2DoE

matrix


lmetho

d

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

Respon

seFOPT

(STB

)15

years

30years

1minus1

11

1minus1

1minus1

minus1

minus1

minus1

2927126833277150

21

minus1

11

minus1

minus1

1minus1

minus1

minus1

2561077628803106

3minus1

11

minus1

minus1

minus1

11

1minus1

3055836634219088

4minus1

minus1

11

1minus1

minus1

11

13063223036143012

51

1minus1

1minus1

minus1

minus1

1minus1

12738475831335538

6minus1

1minus1

11

minus1

1minus1

1minus1

3032718235035264

7minus1

1minus1

minus1

11

minus1

1minus1

minus1

2811745031376578

8minus1

minus1

minus1

1minus1

11

1minus1

12782702832052148

91

11

minus1

1minus1

minus1

minus1

minus1

12648546830122768

10

minus1

minus1

1minus1

11

1minus1

minus1

12707957831057656

11

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

11

2734039031168096

12

1minus1

minus1

minus1

1minus1

11

minus1

minus1

2530320827984070

13

1minus1

1minus1

minus1

1minus1

11

minus1

2682741029853822

14

11

minus1

minus1

minus1

11

minus1

11

2734492031090104

15

1minus1

minus1

11

1minus1

minus1

1minus1

2652291630021116

16

11

11

11

11

11

3013041635331656


Table3PB

desig

ntablefor

10parameters

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

FOPT

(15y

rs)

FOPT

(30y

rs)

1minus1

minus1

minus1

11

1minus1

11

minus1

30848400

33479500

21

minus1

11

minus1

1minus1

minus1

minus1

127154800

29412400

31

11

minus1

11

minus1

1minus1

minus1

28005900

29769600

41

1minus1

11

minus1

1minus1

minus1

minus1

27889000

30076700

5minus1

minus1

11

1minus1

11

minus1

130629500

33815700

6minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

26898400

28727400

71

minus1

1minus1

minus1

minus1

11

1minus1

28034500

29923100

81

minus1

minus1

minus1

11

1minus1

11

27275300

29486200

9minus1

11

minus1

1minus1

minus1

minus1

11

32087300

35060000

10

minus1

1minus1

minus1

minus1

11

1minus1

130096500

32551700

11

minus1

11

1minus1

11

minus1

1minus1

33201400

36053900

12

11

minus1

1minus1

minus1

minus1

11

130582600

33411100




Significance limit

0 5 10 15 20 25 30 35AE

FAHAGADAC

G-MULTFLTK-AQUPV

E-SGCRC-PERMZ

H-KRWD-PORO

B-PERMXJ-ISW

A-OVISC

Contribution ()

Para

met

ers

(a)

0 5 10 15 20 25 30 35

AEF

AHACAGAD



D-POROJ-ISW

A-OVISC

Contribution ()

Para

met

ers

Significance limit

(b)


0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTK-AQUPV

H-KRWC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution

Para

met

ers

Significant

(a)

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTH-KRW

K-AQUPVC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution ()

Para

met

ers

Significant

(b)



0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)




187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10






= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)



= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table3PB

desig

ntablefor

10parameters

Run

AO

VISC

BPE

RMX

CPE

RMZ

DP

ORO

ESG

CRFSW

CRGM

ULT

FLT

HK

RWJISW

KAQ

UPV

FOPT

(15y

rs)

FOPT

(30y

rs)

1minus1

minus1

minus1

11

1minus1

11

minus1

30848400

33479500

21

minus1

11

minus1

1minus1

minus1

minus1

127154800

29412400

31

11

minus1

11

minus1

1minus1

minus1

28005900

29769600

41

1minus1

11

minus1

1minus1

minus1

minus1

27889000

30076700

5minus1

minus1

11

1minus1

11

minus1

130629500

33815700

6minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

minus1

26898400

28727400

71

minus1

1minus1

minus1

minus1

11

1minus1

28034500

29923100

81

minus1

minus1

minus1

11

1minus1

11

27275300

29486200

9minus1

11

minus1

1minus1

minus1

minus1

11

32087300

35060000

10

minus1

1minus1

minus1

minus1

11

1minus1

130096500

32551700

11

minus1

11

1minus1

11

minus1

1minus1

33201400

36053900

12

11

minus1

1minus1

minus1

minus1

11

130582600

33411100




Significance limit

0 5 10 15 20 25 30 35AE

FAHAGADAC

G-MULTFLTK-AQUPV

E-SGCRC-PERMZ

H-KRWD-PORO

B-PERMXJ-ISW

A-OVISC

Contribution ()

Para

met

ers

(a)

0 5 10 15 20 25 30 35

AEF

AHACAGAD



D-POROJ-ISW

A-OVISC

Contribution ()

Para

met

ers

Significance limit

(b)


0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTK-AQUPV

H-KRWC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution

Para

met

ers

Significant

(a)

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTH-KRW

K-AQUPVC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution ()

Para

met

ers

Significant

(b)



0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)




187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10






= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)



= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Significance limit

0 5 10 15 20 25 30 35AE

FAHAGADAC

G-MULTFLTK-AQUPV

E-SGCRC-PERMZ

H-KRWD-PORO

B-PERMXJ-ISW

A-OVISC

Contribution ()

Para

met

ers

(a)

0 5 10 15 20 25 30 35

AEF

AHACAGAD



D-POROJ-ISW

A-OVISC

Contribution ()

Para

met

ers

Significance limit

(b)


0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTK-AQUPV

H-KRWC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution

Para

met

ers

Significant

(a)

0 5 10 15 20 25 30 35 40

F-SWCRE-SGCR

G-MULTFLTH-KRW

K-AQUPVC-PERMZ

D-POROB-PERMX

J-ISWA-OVISC

Contribution ()

Para

met

ers

Significant

(b)



0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)




187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10






= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)



= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25

SGCRSWCR

MULTFLTAQUIPV

KRWPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(a)

0 5 10 15 20 25

SWCRMULTFLT

SGCRKRW

AQUIPVPERMZ

POROSWI

PERMXOVISC

Effects ()

Para

met

ers

Significance limit

(b)




187119864 + 12 1 187119864 + 12 10426 lt00001C2

110119864 + 12 1 110119864 + 12 6171 lt00001D2

566119864 + 13 1 566119864 + 13 316024 lt00001E2

363119864 + 12 1 363119864 + 12 20282 lt00001Residual 644119864 + 11 36 179119864 + 10

Lack of fit 644119864 + 11 31 208119864 + 10






= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

+ 119886

7PERMZ2 + 119886

8SWC2 + 119886

9OVISC2

(4)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

6PERMX2

(5)


= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

8SWC2

+ 119886

9OVISC2 + 119886

10PORO lowastOVISC

(6)



= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











= 119886

0+ 119886

1PORO + 119886

2PERMX + 119886

3PERMZ

+ 119886

4SWC + 119886

5OVISC + 119886

10PORO lowastOVISC

+ 119886

11PORO lowast PERMX

(7)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2

(8)

FOPTFFD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

5PERMX2 + 119887

6SWC2

+ 119887

7OVISC2 + 119887

8PORO lowast PERMX

+ 119887


10PORO lowastOVISC

(9)


= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(10)

FOPTCCD (MMSTB)

= 119887

0+ 119887

1PORO + 119887

2PERMX + 119887

3SWC

+ 119887

4OVISC + 119887

6SWC2

(11)

FOPTD-OPT (MMSTB)

= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2

(12)


= 119862

0+ 119862

1PORO + 119862

2PERMX + 119862

3PERMZ

+ 119862

4AQUIPV + 119862

5SWI + 119862

6KRW

+ 119862

7OVISC + 119862

8SWI2 + 119862

9PERMX2

+ 119862

10PERMZ2 + 119862

11OVISC2 + 119862

12SWI lowastOVISC

(13)

3 Model Validation



119886


0minus621 1697 minus3165 1733

119886

1993 738 2011 1892

119886

2916 3081 289 021

119886

3minus004 020 024 021

119886

425137 806 21460 816

119886


119886

6minus3339 minus1497 000 000

119886

70045 000 000 000

119886

8minus15673 000 minus13286 000

119886

96204 000 3020 000

119886

10000 000 minus1127 minus1270

119886

11000 000 000 260


119887



119887

1977 787 1003 1301

119887

2896 259 290 611

119887

324865 17954 23551 24223

119887


119887


119887

6minus15502 000 000 minus15553

119887

75990 000 000 5845

119887

8000 000 000 257

119887

9000 000 000 7052

119887

10000 000 000 minus1171


119862


119862

03400 3429

119862

1087 090

119862

2113 098

119862

3065 057

119862

4047 059

119862

5111 115

119862

6042 037

119862

7minus150 minus143

119862

8minus058 minus235

119862

9031 000

119862

10minus242 000

119862

11078 000

119862

12minus038 000





Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Pred

icte

d va

lue (

MM

STB)


3E + 07

3E + 07

3E + 07

3E + 07

3E + 073E + 07

4E + 07

4E + 07

4E + 07

4E + 074E + 07



(a)

Pred

icte

d va

lue (

MM

STB)

3E + 07

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 07 4E + 07

4E + 07




(b)

Pred

icte

d va

lues

(MM

STB)

3E + 07

3E + 07

3E + 07 3E + 07

3E + 07

3E + 07

4E + 07

4E + 07

4E + 074E + 07

4E + 07




(c)









119873

sum

119894=1

(Pred minus Exp)


119873

119873

sum

119868=1

1003816

1003816

1003816

1003816

(Pred minus Exp)100381610038161003816

1003816


119873

119873

sum

119894=1



AAPRE = 1119873

[

119873

sum

119894=1

1003816

1003816

1003816

1003816

119864

119894

1003816

1003816

1003816

1003816

]


1003816

119864

119894

1003816

1003816

1003816

1003816

and Min 100381610038161003816

1003816

119864

119894

1003816

1003816

1003816

1003816

119864

119894=

Pred minus ExpExp

lowast 100


119889minus 1

lowast

119873

sum

119894=1

119864

119894

2



0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(a)

0

10

20

30

40

50

60

70

80

30 50 70

Cum

pro

duct

ion

(MM

STB)



(b)

30

32

34

36

38

40

42

44

Cum

pro

duct

ion

(MM

STB)



(c)

10000 12500 15000 17500 20000 22500

Expe

cted

pro

duct

ion

(STB

)

Days


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07

5E + 07

(d)




AAD 845652 1413333 1533333 2276923


119864max 09 15 14 21

SD 01 03 03 08

AAPRE 03 04 05 07

119877-square () 996 994 993 979

Adj 119877-square () 995 992 989 973

Pred 119877-square () 994 988 982 957

Exp runs 460 320 310 470





AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












AAD 907143 941667 1841667 4485714


119864max 08 11 15 51

SD 02 02 05 35

AAPRE 03 03 06 14

119877-square () 996 997 991 924

Adj 119877-square () 995 996 989 899

Pred 119877-square () 992 995 985 844

Exp runs 290 840 240 250





















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















0

01

02

03

04

05

06

07

08

09

1

25 30 35 40 45 50

Perc

entil

e

Reserves (MMSTB)



384MMSTB36MMSTB

354MMSTB








8 Summary








0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0010203040506070809

1

28 30 32 34 36 38 40 42 44

Perc

entil

es

Reserves (MMSTB)


(a)

0010203040506070809

1

20 25 30 35 40 45 50

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(b)

0010203040506070809

1

29 31 33 35 37 39

Perc

entil

es

Reserves (MMSTB)

FOPT_FRFDFOPT_PBD

(c)








0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

005

01

015

02

025

03

Prob

abili

ty d

ensit

y

Reserves

26 28 30 32 34 36 38 40


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

271905409620340595200213165223655682000

Reserves


(a)

0

002

004

006

008

01

012

Prob

abili

ty d

ensit

y

Reserves

20 25 30 35 40 45 50


00

167

333

500

667

833

1000

Cum

ulat

ive d

ensit

y (

)

291509379377340764134983231033574902000

Reserves


(b)

00

167

333

500

667

833

1000

0

002

004

006

008

01

012Cu

mul

ativ

e den

sity

()

Prob

abili

ty d

ensit

y


Reserves

20 25 30 35 40 45 50

207710459520341152358312958093870282000

Reserves


(c)



029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










029

021

019

0 02 04

PORO

PERMX

PERMZ

SWC

OVISC


minus044

minus064


Coefficient value


12000 14500 17000 19500 22000

Cum

pro

duct

ion

(STB

)

Days

P90

P50P10


3E + 07

2E + 07

2E + 07

3E + 07

4E + 07

4E + 07




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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