Ira. J. Chem. Chem. Eg. Vol. 6, No., 007 Sesitivity Aalysis of Water Floodig Optimizatio by Dyamic Optimizatio Gharesheiklou, Ali Asghar* + ; Mousavi-Dehghai, Sayed Ali Research Istitute of Petroleum Idustry (RIPI), P.O. Box 8745-463 Tehra, I.R. IRAN ABSTRACT: This study cocers the scope to improve water floodig i heterogeeous reservoirs. We used a existig, i-house developed, optimizatio program cosistig of a reservoir simulator i combiatio with a adjoit-based optimal cotrol algorithm. I particular we aimed to examie the scope for optimizatio i a two-dimesioal horizotal reservoir cotaiig a sigle high permeable streak, as a fuctio of reservoir ad fluid parameters, which we combied i the form of 0 dimesioless parameters. We defied the parameter NPV improvemet to idicate the improvemet i et preset value (NPV) that ca be achieved through optimizatio. For iitial screeig of the effect of the dimesioless parameters, a two-level D-optimal desig of experimets (DOE) techique was used to obtai a liear respose surface model with the aid of waterfloodig simulatios. As a result 8 dimesioless groups were selected for more detailed aalysis, ad a full quadratic NPV improvemet model was costructed usig a three-level D-optimal desig usig 50 simulatios. It should be remided that all the D-optimal matrix desigs were geerated by usig commads of statistics Toolbox of MATLAB software. Fially, Pareto charts were plotted to visualize the sesitivity of the model as a fuctio of the dimesioless parameters. Based o the preset model we ca draw the coclusio that the parameters L s / L (relative streak legth), kmax / k (relative streak permeability) ad the ratio of water cost ad oil price have the maxstreak largest effect o the scope for obtaiig a high value of NPV improvemet. KEY WORDS: Water floodig optimizatio, Dyamic optimizatio, Desig of experimets, Multiple liear regressio, Respose surface model, NPV improvemet. INTRODUCTION I the oilfield, like the real world, itelligece is ot always a guaratee for success, ad the key parameter i the developmet of smart well techology is whe the added fuctioality also adds value. Therefore, efficiecy * To whom correspodece should be addressed. + E-mail: shikhlooaa@ripi.ir 0-9986/07//93 3/$/3.30 of differet scearios of smart well techology should be examied before ay practical implemetatio. Water floodig optimizatio usig dyamic optimizatio is oe of these smart well scearios, which aims to maximize 93
Ira. J. Chem. Chem. Eg. Gharesheiklou, A. A., et al. Vol. 6, No., 007 Defie iitial parameters Dimesioal aalysis of parameters Defie objective Fuctio Fittig the liear model by multiple liear regressio Ru simulatios ad calculate NPV improvemet Desig simulatios for liear screeig by DOE Desig respose surface model simulatios by D-optimal DOE Select the dimesioless parameters with the highest effect o NPV improvemet recovery or et preset value over a give time period for heterogoous reservoirs that suffer from high permeable streaks. As a matter of fact, the high permeable streaks ofte cause early breakthrough; hece ijected water escapes through the high permeable streaks ad oil sweepig remais immature. However, dyamic flow cotrol geerally helps to solve this problem cosiderably [,3-5]. A average improvemet of 3.7 % was see i the 50 simulatio rus. It was kow that the scope for optimizatio by dyamic flow cotrol depeds o reservoir properties such as legth ad width of the high permeable streaks []. Therefore, we aimed to examie the sesitivity of this scope for improvemet with respect to various reservoir properties by costructig a respose surface model. This report documets the efforts aimed to achieve the goal of the project. Fig. : Layout of the project. LAYOUT OF THE RESEARCH The complete layout of the project has bee described schematically i Fig.. Accordig to table, we started with selectio of iitial parameters. The we chaged iput variables to dimesioless parameters to reduce their umber ad elimiate their dimesios (table ). I other words, all the iput variables were coverted to dimesioless parameters by costructio of the Ru simulatios ad calculate NPV improvemet Iterpret the respose surface model Fittig the respose surface model by multiple liear regressio Error aalysis ad idepedet testig of the respose surface model dimesioless ratios. We aimed to examie the sesitivity of output fuctio of the optimizer (NPV) i terms of the reservoir ad fluid properties by costructig a respose surface model. Therefore, the respose fuctio (NPV improvemet ) was defied o the basis of NPV fuctio ad dimesioless variables were cosidered as iputs of the respose surface model. D-optimal techique was used to desig the screeig simulatio rus. After ruig the simulatios, multiple liear regressio was cosidered to fit a liear model so that we could scree the major effects of dimesioless parameters o respose fuctio. Pareto charts were costructed to visualize the result of liear screeig. After elimiatig two iitial dimesioless parameters (µ o /µ w, p res /p ref ) with small effects o respose fuctio, fial simulatio rus were desiged by usig of three-level D-optimal techique. By ruig the simulatios ad recordig the NPV improvemet for all the rus, full-quadratic respose surface model was costructed by usig multiple liear regressio. Pareto charts were plotted i order to depict the effects of liear iteractio ad squared terms of the respose surface model. The ext step was idepedet testig ad error aalysis of the model. It was doe i order to examie the efficiecy of the costructed respose surface model. 94
Ira. J. Chem. Chem. Eg. Sesitivity Aalysis of Water Vol. 6, No., 007 Iitial variable kx streak ky streak kx matrix ky matrix Table : Iitial iput parameters. Descriptio Streak permeability i x-directio Streak permeability i y-directio Matrix permeability i y-directio Matrix permeability i y-directio µ o Oil viscosity µ w Water viscosity p r q ij L s L W s W Water cost Oil price ϕ Reservoir pressure Water ijectio rate Legth of streak Legth of reservoir Width of streak Width of reservoir Water productio cost Oil price Porosity Table : Iitial dimesioless parameters. Dimesioless group ϕ k max /k maxstreak A streak A matrix Descriptio Porosity Maximum permeability of matrix over maximum permeability of streak Aisotropy of streak Aisotropy of matrix µ o /µ w Oil viscosity over water viscosity W s /W L s /L p res /p ref Q ij /q ref (Water cost/oil price) Width of streak over width of reservoir Legth of streak over legth of reservoir Reservoir pressure over referece pressure Water ijectio rate over referece water ijectio rate Water produced cost over oil price CHARACTERISTICS OF THE OPTIMIZER Reservoir model descriptio Water floodig optimizer assumes a heterogeeous, horizotal, two dimesioal, two-phase (oil-water) reservoir with two horizotal smart wells, a ijector ad a producer, at opposite sides. The reservoir has o-flow boudaries at all sides. Each well is divided i segmets with ICVs, allowig for idividual iflow cotrol of the segmets [3]. Dimesios of the grid blocks are 30 m (legth), 30m (width) ad 0m (height). There are 30 grid blocks alog the legth ad width of the reservoir. Therefore, the area of the reservoir is 900 m by 900 m. High permeable streaks ca be defied i twodimesioal reservoir model with differet legths, widths ad agle. The streaks cause early water breakthrough, ad water floodig optimizer aims to maximize recovery or et preset value over a give simulatio time for heterogoous reservoirs that suffer from these high permeable streaks. Fig. ad Fig. 3 describe the heterogeeous, horizotal, two-dimesioal reservoir. Ijectio ad productio wells are located at opposite sides of the reservoir ad high permeable streaks are perpedicular to the directio of ijectio ad productio wells. Reservoir model icludes two differet parts: streak ad matrix. They both have the same porosity, but differet permeability i x ad y-directio. Streaks with differet legths ad widths ca be defied for optimizer i order to moitor the effect of chage of them o output fuctio of the optimizer. Methodology of the water floodig optimizer - Water floodig optimizer has bee developed o the basis of optimal cotrol algorithm. Optimal cotrol is a gradiet-based optimizatio techique that is used to fid the iput variables that miimize or maximize a certai objective fuctio. But maximizig or miimizig of the objective fuctio is uder limitatio of differet costraits of the system. Dyamic optimizatio ivolves the costraits of the system by usig of Lagrage Multipliers [3]. Water floodig optimizer has cosidered two differet optios for reservoir costraits: - Pressure costrait - Rate costrait We ru all the desiged simulatios uder the assumptio of rate costrait. 95
Ira. J. Chem. Chem. Eg. Gharesheiklou, A. A., et al. Vol. 6, No., 007 30-grid block alog the width of the reservoir Fig. : Visualizatio of the dimesios of reservoir ad a selective high permeable streak. (Color bar shows the differece of permeability of matrix ad high permeable streak). DEFINING INPUT PARAMETERS AND OUTPUT RESPONSE FUNCTION Iitial iput parameters Table itroduces differet iput variables of the optimizer. Two-dimesioal reservoir model assumes four differet permeabilities (matrix permeability alog x ad y-directio, streak permeability alog x ad y- directio). Oil ad water viscosity were the ext iput variables. Geometric dimesios of the high permeable streak ad reservoir (i.e. legth ad width of the reservoir ad streak) were the other iput variables. Reservoir pressure, water ijectio rate, water productio cost, oil price ad reservoir porosity were the last selectios for iitial iput parameters. Of course, there were some other variables such as capillary pressure (P c ), oil ad water compressibility (c o, c w ), ad agle of streak, but their small effects had already bee cofirmed by early simulatio rus. Output respose fuctio NPV has bee cosidered as objective fuctio, which teds to be maximized by optimal cotrol algorithm [3]. The objective fuctio is equal to the NPV: ς = where 5 0 5 0 5 30 N ς m= 0 High permeable streak with 4-grid block legth ad 3-grid block width 5 0 5 0 5 30 30-grid block alog the reservoir legth 9 8 7 6 5 4 3 () Directio of horizotal ijectio well 5 0 5 0 5 30 Width of streak Legth of streak 5 0 5 0 5 30 30-grid block alog the reservoir legth Fig. 3: Top view of reservoir showig two-dimesioal permeability of matrix ad streak. Nprod rw (q w ) k ro (qo ) k ς = x yh t K t = () ( + b /00) The assumptio is that the grid block volume x yh ad the cost/beefit coefficiets r w, ad r o are the same for all well segmets. The terms i the deomiators of the sum terms itroduce discoutig. ς is the et preset value fuctio over a give time, b the aual iterest rate which is expressed i %, t the time expressed i whole years at time step,q w produced water flow rate ad q o produced oil flow rate. Note that q o ad q w have egative sig ad t is a expoet, where is a superscript [3]. Geerally, high permeable streaks cause early water breakthrough ad water-floodig optimizer has bee desiged to maximize recovery i heterogeeous reservoirs. The optimizer improves the water floodig policy from differet segmets with iteratio process. First of all, the optimizer performs water floodig without the use of optimizer algorithm (i.e. covetioal water floodig). After that, the water ijectio policy proceeds o the basis of optimal cotrol algorithm i order to maximize the et preset of the water-floodig sceario. Therefore, optimizer calculates et preset value of the water floodig scearios with ad without usig optimizatio algorithm. They were amed as NPV optimized ad NPV covetioal, respectively. We defied NPV improvemet 9 8 7 6 5 4 3 Directio of horizotal productio well 96
Ira. J. Chem. Chem. Eg. Sesitivity Aalysis of Water Vol. 6, No., 007 as respose fuctio i order to moitor the efficiecy of the optimizer algorithm i differet reservoir coditios: NPV improvemet NPVoptimized NPVcovetioal = (3) NPV covetioal where NPV covetioal is et preset value without the use of water floodig optimizer, NPV optimized et preset value by meas of optimizer algorithm ad NPV improvemet the fractio of et preset value improvemet. Costat cumulative water ijectio We cosidered oe pore volume ijectio for all the simulatio rus. I this case, we were sure that all the simulatio rus reach to the breakthrough time ad the reservoir is ot depleted completely. We calculated simulatio time for differet rus by usig the followig formula: Simulatio time= (pore volume) /( ijectio rate) (4) Desig of experimets (DOEs) Sometimes we wat to kow the sesitivity of output fuctio of a system whit respect to the differet iput parameters. The system ca be experimetal set-up or simulatio software. Therefore, we deliberately chage oe or more iput parameters to observe the effects of the imposed perturbatio o output fuctio. Oe by oe chagig of the parameters is the traditioal way of perturbig the iput parameters. I this case, we have to chage the iput variables oe by oe ad ru the simulatio or experimetal set-up for all the parameter chages. This is ot a efficiet way, because it caot cosider the simultaeous effects of chages of differet parameters ad it is also very time- cosumig. The statistical desig of experimets (DOE) is a amuch more improved procedure for plaig experimets so that data ca be aalyzed to give valid coclusios. DOE itroduces differet techiques i order to moitor the simultaeous chages of iput parameters i a systematic way [6]. The techique is applied to choose a moderate umber of simulatio rus ad aalyze them to estimate the sesitivity of output fuctio to various iput parameters. I other words, well-chose experimetal desigs ca maximize the amout of iformatio that ca be obtaied for a give amout of experimetal desig [6]. I order to costruct a certai DOE desig for simulator or experimetal set-up, there was a eed to defie levels of extremes for each iput variable. Twolevel (i.e. miimum ad maximum) ad three-level (i.e. miimum, itermediate, ad maximum) desigs are the most prevalet oes. Two-level desigs caot be used to predict the curvature shape of respose surfaces. They oly ca costruct liear respose surfaces, whereas three-level desigs ca be used i order to costruct oliear respose surface models. I liear screeig of the iitial dimesioless parameters, we defied two levels of extremes for each dimesioless parameter ad we reached to the third itermediate level by averagig of maximum ad miimum levels. Therefore, the three levels were used for costructio of three-level D-optimal desig. For example, porosity of the reservoir was oe of the iput dimesioless parameters. We cosidered 0. ad 0.3 for miimum ad maximum levels, respectively. They were used for liear screeig. After screeig out the dimesioless parameters, porosity was still oe of the fial dimesioless parameters. We eed to have three levels for each fial dimesioless parameter. Therefore, we obtaied 0. for itermediate level of reservoir porosity by averagig of 0. ad 0.3. The the three levels were used for creatig o-liear respose surface model. Two levels of extremes were defied to make liear model ad three levels of extremes were cosidered to create the oliear respose surface model. Note that otatios, 0, + describe miimum, itermediate ad maximum levels of parameters, respectively. I geeral, desigs are lists of combiatio of factors at which experimets or simulatios are performed. I matrix otatio, each row of the desig matrix idicates a ru, whereas each colum cotais the settigs of each factor [6]. Table 3 ad table 4 show the extreme levels of dimesioless parameters used for costructig the liear ad oliear respose surface models. Full factorial desigs are the simplest forms of DOE desigs ad the umber of simulatio rus for full factorial desigs ca be calculated by the followig formula: Number of full factorial rus= L k (5) where L is the umber of levels of factors ad k the 97
Ira. J. Chem. Chem. Eg. Gharesheiklou, A. A., et al. Vol. 6, No., 007 Table 3: Iitial dimesioless parameters ad their two-level extremes for liear screeig. Dimesioless parameter Low extreme High extreme φ 0.0 0.30 k max /k max streak 0.0 0.0 A streak 0.0.00 A matrix 0.0.00 - Classical experimetal desigs - Optimal experimetal desig The mai differece betwee classical ad optimal experimetal desigs is that classical desigs are the oes created before the geeratio of computers, but optimal desigs were developed after ivetio of computers. Therefore, classical desigs are famous to first-geeratio of DOE desigs ad optimal experimetal desigs are called secod-geeratio of DOE desigs [7]. µ o /µ w 0.0 0.00 W s / W 0.03 0.33 L s / L 0.80.00 P res / p ref 0.75.00 q ij / q ref 0.60.00 (Water cost)/(oil price) 0.05 0.0 Table 4: Levels of extremes for 3-level D-optimal desig. Dimesioless parameter Low extreme Itermediate level High extreme φ 0. 0. 0.3 k max /k maxtreak 0.0 0.055 0. A streak 0.0 0.505 A matrix 0.0 0.505 W s /W 0.06 0.8 0.3 L s /L 0.8 0.9 Q ij /q ref 0.6 0.8 (Water cost) / (Oil price) 0.05 0.5 0. umber of iput parameters. The story of DOE desig begis from full factorial desig. Needless to say, full factorial desigs iclude all the possible settigs ad are the most complete desigs. But if we have much more iput variables with three or more levels of extremes, the umber of simulatio rus will icrease dramatically. Therefore, differet techiques of DOE were iveted i order to have moderate umber of simulatio rus with high amout of data iformatio. Differet techiques of DOE ca be divided ito two mai groups: Optimal experimetal desig Optimal experimetal desigs are called secodgeeratio of DOE desigs sice they were developed after geeratio of computers. They are all based o mathematical optimality criterio. Hece, usig of computer is ievitable for costructig optimal experimetal desigs. D-optimal DOE desigs are the most importat types of optimal experimetal desigs. D-optimal desig The mai idea of the D-optimal DOE desig ca be described by the followig formula: (Iformatio) /(Simulatio rus) = maximum. (6) I other words, D-optimal desig helps to desig simulatio rus with the maximum amout of iformatio ad miimum umber of simulatio rus. D-optimal desig is based o the followig optimality criterio: Two colum vectors X ad X are orthogoal if X * X =0.The more depedet the vectors (colums) of the desig matrix, the closer to zero is the determiat of the correlatio matrix for those vectors; the more idepedet the colums, the larger is the determiat of that matrix. Therefore, fidig a desig matrix that maximized the determiat D of the desig matrix meas fidig a desig where the factor effects are maximally idepedet of each other. This criterio for selectig a desig is called D-optimality criterio [8]. We used two-level D-optimal desig for costructig of liear matrix desig. We geerated simulatios rus for 0 iitial dimesioless parameters. After that, threelevel D-optimal desig was used i order to create 50 simulatio rus for eight dimesioless parameter. All the D-optimal matrix desigs were geerated by usig of commads of statistics Toolbox of MATLAB software. Shape of the model (i.e. liear, iteractio- liear, 98
Ira. J. Chem. Chem. Eg. Sesitivity Aalysis of Water Vol. 6, No., 007 Defie the umber of factors, umber of levels for each factor ad the levels of extremes Costruct matrix desig by DOE techiques such as D- optimal, fractioal factorial etc Put each row i simulator, ru the simulator ad obtai the actual output fuctio Fit the model by multiple liear regressio quadratic), umber of desired rus ad umber of parameters were three essetial iput terms for costructio of D-optimal matrix desigs [9]. We used D-optimal desigs for liear ad o-liear modelig because they have bee costructed o the basis of optimality criterio. Namely, they have maximum amout of iformatio with miimum amouts of simulatio rus. Therefore, they are always good cadidates for makig matrix desigs. But the poit is that D-optimal desigs are created uder the assumptio of shape of respose surface fuctio. Therefore, D- optimal desigs are always biased o the assumed shape of respose surface model. The complete procedure for costructio of respose surface models usig desig of experimets method has bee described i Fig. 4. RESPONSE SURFACE MODELS Geerally, the true relatio betwee the differet parameters of a system is really ukow. Therefore, fidig a approximate solutio as a empirical equatio is a good way to predict the effect of iput idepedet parameters o output depedet fuctio. Respose surface models are fuctios that are empirically fit to the observed data from results of experimets or simulatio rus. They cosider a polyomial empirical equatio to predict the local shape of the respose surface. That s why they are amed respose surface models. Polyomial respose surface models are most widely used so that we ca fid optimum or improved process settigs [0]. Liear ad quadratic respose surface models are two Fig. 4: DOE process for costructig of respose surface models. Substitute miimum, itermediate ad maximum levels of each factor i place of,0,+ respectively i a fixed colum importat kids of polyomial respose surface models. Besides, iteractio terms ca be added to empirical structure of respose surface models. Therefore, there are differet optios i order to desig the model equatio. They are as follows: - Costat term - Liear term - Iteractio term - Quadratic term For istace, if we cosider a system with two iput variables (x,x ), full quadratic respose surface model ca be described by the followig formula []: 0 + βx + βx + βxx + βx + βx η = β (7) As we see i the above formula, full quadratic respose surface model, icludes costat term (β 0 ), liear terms (β x +β x ), iteractio terms (β x x ) ad quadratic terms ( β x + β x ), where β 0, β, β, β, β, β are correspodet coefficiets of differet terms of the equatio ad η is the model fuctio. It is possible to costruct the model fuctio with differet structures by the use of a combiatio of costat term, liear term(s), iteractio term(s) ad quadratic term(s) []. The followig formulas describe some other optios of respose surface model with two iput parameters (x, x ): Simple liear model: η=β 0 +β x +β x (8) Liear-iteractio model: 99
Ira. J. Chem. Chem. Eg. Gharesheiklou, A. A., et al. Vol. 6, No., 007 η = β + β x + β x + β x x (9) o The otatios used for the above formulas are the same as full quadratic respose surface equatio. Respose surface models ca be used as a proxy model for reservoir simulators i order to perform ucertaity aalysis, parameters estimatio ad optimizatio. Fittig the experimetal desig by multiple liear regressio We used multiple liear regressio i order to fit the liear ad o-liear respose surface models. Multiple liear regressio is a statistical techique that allows us to predict oe depedet variable with respect to several idepedet variables [3]. Multiple liear regressio is multidimesioal liear regressio o the basis of Least square method. Least square method assumes that the best curve-fit of data is the curve that has the miimal sum of the deviatios squared (least square error) from a give series of data. Suppose that the data poits are: ( x, y),(x, y,..., x, y ) (0) Where x is idepedet variable ad y is the depedet variable. The fittig curve f(x) has the deviatio (error) d from each data poit. Accordig to the method of least squares, the best fittig curve should satisfy the followig formula: i= d i = i i = i= [ y f (x )] Miimum () Geerally, the purpose of multiple liear regressio is to fid a relatioship betwee a group of iput parameters (the colums of x) ad a respose, y. This relatioship is useful for uderstadig which parameters have the greatest effect, kowig the directio of the effect (i.e., icreasig x icreases/decreases y) ad usig the model to predict future values of the respose whe oly the predictors are curretly kow [3]. The liear model ca be described by the followig formula: y = Xβ + ε () where y is a -by- vector of observatios, X a -by-p matrix of repressors, β a p-by- vector of parameters ad ε a -by- vector of radom disturbaces. The solutio to the problem is a vector, b, which estimates the ukow vector of parameters, β. The least squares solutio is: T T b = βˆ = (X X) X y () Geerally, the purpose of multiple liear regressio is to fid a relatioship betwee a group of iput parameters (the colums of x) ad a respose, y. This relatioship is useful for uderstadig which parameters have the greatest effect, kowig the directio of the effect (i.e., icreasig x icreases/decreases y) ad usig the model to predict future values of the respose whe oly the predictors are curretly kow [3]. The liear model ca be described by the followig formula: y = Xβ + ε () where y is a -by- vector of observatios, X a -by-p matrix of repressors, β a p-by- vector of parameters ad ε a -by- vector of radom disturbaces. The solutio to the problem is a vector, b, which estimates the ukow vector of parameters, β. The least squares solutio is: ˆ T T b = β = (X X) X y (3) We used multiple liear regressio i order to solve the combiatio of simulatio desig equatio. All the liear ad oliear models were fitted by usig of commads of Statistics Toolbox of MATLAB software. Liear screeig of the parameters The primary purpose of the liear screeig is to select or scree out the few importat mai effects from the may less importat effects. Liear screeig desigs are also amed mai effects desigs. The methodology of liear screeig applies solvig a liear program with the regressio fuctio, estimated through liear regressio aalysis, as the objective fuctio. Liear screeig assumes that iput parameters are completely idepedet of each other. Therefore, it is ot a real respose surface model. It is usually doe to scree the major effects of the iitial iput parameters. Accordig to the results of liear screeig, uecessary iitial parameters with the least degrees of effects will be elimiated. Therefore, fial oliear respose surface model is costructed o the basis of the most importat iput parameters. We selected 0 differet dimesioless 00
Ira. J. Chem. Chem. Eg. Sesitivity Aalysis of Water Vol. 6, No., 007 K max /K max streak Porosity Water cost$/oil price$.3 A -0.979 LS/L 0.646 Pr/Pref 0.634 WS/W -0.4864 Astreak -0.49 Qij/qref -0.3308 Muo/muw -0.0684 0 3 4 5 Fig. 5: Pareto chart showig liear model coefficiets. groups ad cosidered a simple liear model for liear screeig: 0 N PˆV improveme = β + β x (4) t 0 Coefficiets of liear model i= i where x i ad β i represet the dimesioless groups ad their correspodet coefficiets, respectively. We cosidered two extremes for each dimesioless group ad substituted them i a two-level D-optimal desig ad cosidered a simple liear model (for 0 iitial dimesioless parameters) i order to fit simulatio rus of screeig part by multiple liear regressio. Fially we derived the liear model coefficiets by usig multiple liear regressio techique. Pareto graphs were costructed to visualize the results of screeig part.see Fig. 5 ad Fig. 6. It should be remided that Pareto Chart is a special form of a bar graph ad is widely used to display the relative importace of problems or coditios. Pareto charts graphically order the effects of factors with respect to the respose fuctio so that the most importat effects (mai effects) ca come to the surface. I geeral, a Pareto chart is used for focusig o critical issues by orderig them i terms of importace ad frequecy, prioritizig problems ad aalyzig problems by differet groupigs of data. i.88-4.03 Costructio of respose surface model Geerally, the structure of the relatioship betwee the respose ad the idepedet variables is ukow. Water cost$/oil price$ 6.99 % A 7.80 % LS/L 5.5 % Pr/Pref 5.05 % WS/W 3.87 % Astreak 3.4 % Qij/qref.63 % Muo/muw 0.54 % 0 0 0 30 40 50 Fig. 6: Pareto chart showig ormalized effects of dimesioless groups i liear model. The most importat step i the costructio of respose surface model is to fid a suitable approximatio to the real relatioship. The most commo forms of respose surface models are low-order polyomials (first or secod-order). First-order models are widely used for screeig of the iitial iput parameters ad secod-order respose surface models are the most commo forms of oliear respose surface models. They are also famous to the full-quadratic respose surface models. They iclude liear, square ad iteractio terms. The secodorder model, i geeral form, is give as: η = β K max /K max streak Porosity 0 + jj j ij i x j β x + β x (5) where η is full-quadratic respose surface model, β o the costat term, β j x j liear terms, β jj x j square terms ad β Normalized effects of dimesioless groups ij x x iteractio terms. Note that i j 3.0 %.44 % β 0, β j, β jj ad β ij are correspodet coefficiets of differet terms. The major effects of iitial dimesioless parameters had already bee moitored by liear screeig. Referrig to the liear screeig results, we elimiated two dimesioless parameters (µ o /µ w, p res /p ref ) with small effects o output respose fuctio ad selected eight fial dimesioless parameters. We costructed a fullquadratic respose surface model with eight dimesioless parameters. A full-quadratic respose surface model with eight variables icludes oe costat term, 0
Ira. J. Chem. Chem. Eg. Gharesheiklou, A. A., et al. Vol. 6, No., 007 eight liear terms (mai effects of each iput parameter), eight square terms (square terms of each iput parameter) ad 8 iteractio terms (all the bilateral effects of the iput parameters). Totally costructed full-quadratic respose surface model icluded 45 terms. We desiged 50 simulatio rus by usig three-level D-optimal techique of DOEs for eight parameters. Afterruig the simulatios ad calculatig NPV improvemet for all the rus, we obtaied the 50 by solutio matrix (y). The we completed 50 by 45 iformatio matrix (X) by multiplyig ad squarig the correspodet pre-desiged mai effects. Usig multiple liear regressio, we obtaied 45 by coefficiet matrix (β) of full-quadratic respose surface model. Fially Pareto charts were costructed i order to visualize the effects of 45 terms of full-quadratic respose surface model. See Fig. 7. Idepedet testig of respose surface model by perturbatio of DOE desig Respose surface models are always biased to the levels of desigs. Namely, they are costructed by the extremes of the iput parameters. O the other had, the structure of the relatioship betwee the respose fuctio ad iput parameters is ukow ad costructio of respose surface models is doe o the basis of fidig a suitable approximatio to the actual relatioship. Hece, respose models are always biased o levels of the iput parameters ad the pre-assumed structure of the model. Therefore, we perturbated the DOE desig by geeratig te ew simulatio desigs to see the efficiecy of the model for predictig the simulatio rus. The perturbig desig was geerated by defiig ew extremes o the basis of liear model assumptio. It was doe i order to impose chages to the mai DOE desig to see the effects of the perturbatio o output fuctio of the model. We defied the perturbig rus by two-level D-optimal DOE techique ad calculated actual NPV improvemet for each ru. O the other had, we calculated NPV improvemet by usig of the oliear surface respose model. Compariso of the actual ad calculated NPV was doe i order to examie the efficiecy of the costructed respose surface model i predictig the output respose fuctio, Fig. 8 depicts the ability of the model for predictig the recovery improvemet scearios. Error aalysis was doe i order Fig. 7: Pareto chart showig the ormalized effects of oliear respose surface model. 0.0 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.0 0.0 0.00 0.00 0.0 0.0 0.03 0.04 0.05 0.06 NPV actual NPV model G Cl I G CG BI CF A CH B BC C BG F B BH I BF FH F FG DI H EI GI EG D DG E H BD GH CD C DH EH EF DF HI CE D DE E FI BE.34 %.5 %.9 %.5 %.0 %.05 % 0.66 % 0.65 % 0.64 % 0.55 % 0.5 % 0.5 % 0.50 % 0.47 % 0.45 % 0.45 % 0.36 0.9 % 0. % 0.0 % 0.8 % 0.5 % 0. % 0.09 % 0.07 % 0.06 % 0.045 %.84 %.7 %.3 %.0 %.0 %.69 %.45 %.39 %.35 % 7.6 % 6.8 % 6.65 % 6.6 % 5.97 % 5.4 % 8.9 % 8.74 %.99 % 0.00.00 4.00 6.00 8.00 0.00.00 4.00 Percetage effect Fig. 8: Scattered plot showig five poits of idepedet testig. to examie the efficiecy of the proposed respose surface model. Table 5 ad table 6 show the results of error aalysis i detail. RESULTS AND DISCUSSION Liear screeig Liear screeig always is doe to detect the iput parameters with the major degrees of importace. A B C D E F G H I Costat term Porosity K max / K maxstreak Astreak A Ws/W Ls/L qijectio/qref Water cost ($)/Oil price($) 0
Ira. J. Chem. Chem. Eg. Sesitivity Aalysis of Water Vol. 6, No., 007 Table5: selective idepedet testig of the model. Rus NPV actual NPV model 0.0 0.046 0.0485 0.089 3 0.0 0.057 4 0.007 0.0046 5 0.03 0.0339 Table 6: The results of error aalysis. Mea absolute error 0.063 Mea NPV actual 0.05 Mea abs error/mea NPV actual 0.75836 Namely, liear screeig is a rough estimate for moitorig the sesitivity of the output respose fuctio with respect to the differet iput parameters. It does t cover iteractio ad secod order terms. Therefore, liear model is ot a real respose surface model ad is uable to pipoit the effects of differet parameters exactly. However, referrig to the Pareto charts that show the screeig results of iitial dimesioless parameters (Fig. 5 ad Fig. 6), k max /k max has the highest effect o output respose fuctio. O the other had, µ o /µ w has the least importace ad the effects of the other dimesioless parameters are betwee these to extreme effects. Geometric factors (W s /W, L s /L) with moderate effects are i the middle of the order of the Pareto chart. Full quadratic respose surface model Accordig to the o-liear respose surface model (Fig. 7) mai effect G (legth of streak over legth of reservoir) has the maximum effect o output respose fuctio. Square term G ad iteractio term CG also have high effects o respose fuctio. Therefore, it is safe to say that legth of streak over legth of reservoir has the highest effect o output fuctio. Iteractio term Cl ad square term I are the secod ad third terms i Pareto chart rakig. As a result, fiacial dimesioless group I (water cost over oil price) is the secod importat dimesioless group. Referrig to the positio of iteractio terms Cl, CG ad CF, oe ca fid out that C(k max /k maxstreak) ) is the third importat dimesioless parameter. The other terms have the lower degrees of effect. It ca be cocluded that iteractio ad square terms have importat effects o respose surface fuctio. Therefore, they affect the rakig of the differet terms ad this is the mai reaso for the differece betwee the results of liear screeig ad o-liear surface respose model. Legth of streak ad variatio of matrix ad high permeable streak permeability both itesify early water breakthrough. Surprisigly, the effect of porosity o NPV improvemet is quite cosiderable, whereas we expected that porosity could t play a importat role o NPV improvemet. However, the model should be improved (i.e. by icreasig simulatio rus), i order to judge o the importace of the other terms at the bottom of the Pareto graph. Idepedet testig of the respose surface model A umber of 0 idepedet rus were desiged i order to examie the fittig efficiecy of the model. Table 6 shows the results of idepedet testig of the model. As we see, respose surface model has predicted simulatio rus with a acceptable amout of error. Fig. 8 depicts the scattered graph of idepedet testig results. It shows that respose surface model is successful to predict these simulatio rus. It ca be imagied that there is a parabolic tred betwee the umber of simulatio rus ad fittig efficiecy of the respose surface model ad that by icreasig simulatio rus, fittig efficiecy of the model will icrease dramatically, but after that, the slope will be smoothed gradually. However, icreasig simulatio rus is highly recommeded to complete the respose surface model as a fast proxy model. Recommedatios - Icrease the umber of simulatio rus to complete the respose surface model. - Try other optios for alterative respose surface models, i.e. cubic or logarithmic models. CONCLUSIONS - The scope to improve water floodig i a twodimesioal square reservoir produced with two rows of ijectio ad productio wells ad cotaiig a sigle heterogeeous streak ca be described with the aid of 0 dimesioless parameters. 03
Ira. J. Chem. Chem. Eg. Gharesheiklou, A. A., et al. Vol. 6, No., 007 - Iitial screeig usig a liear respose model based o water-floodig simulatios idicated that oly 8 dimesioless parameters are statistically sigificat. 3- More complicated respose surface models ca be easily obtaied with the aid of desig-of-experimets techiques. I particular, we developed a full quadratic respose surface model based o 50 simulatios. 4- These simulatio rus show a average scope of improvemet of 3.7 %. 5- A idepedet verificatio of the quality of the full quadratic respose surface model, usig aother 0 simulatios, reveals that some more simulatios is eeded to obtai a more reliable respose surface model. 6- Based o the preset model we coclude that the parameters L s /L (relative streak legth), k max /k maxstreak (relative streak permeability) ad the ratio of water cost ad oil price have the largest effect o the scope to improve water floodig. The scope for improvemet icreases with larger relative streak legth, larger permeability cotrast, ad relatively high water costs. Nomeclatures NPV Net preset value NPV improvemet Net preset value improvemet NPV covetioal Net preset value without optimizer NPV optimized Net preset value with optimizer Time step N Fial time step r Price per uit volume,($/m 3 ) t Time iterval, (s) q o Oil flow rate, (m 3 /s) q w Water flow rate, (m 3 /s) b Aual iterest rate h Reservoir height, (m) k maxstreak Maximum permeability of streak, (m ) k max Maximum permeability of matrix, (m ) L Legth of reservoir, (m) L s Legth of streak, (m) W s Width of reservoir, (m) W Width of streak, (m) A Matrix aisotropy A streak Streak aisotropy ϕ Porosity µ o Oil viscosity, (Pa.s) µ w Water viscosity, (Pa.s) p res p ref q ij q ref Reservoir pressure, (Pa) Referece pressure, (Pa) Ijectio rate, (m 3 /s) Referece ijectio rate, (m 3 /s) ky matrix Matrix permeability i y directio, (m ) ky streak Streak permeability i y directio, (m ) ky streak Streak permeability i y directio, (m ) kx matrix Matrix permeability i x directio, (m ) RSM c o c w p c ICV Respose surface model Oil compressibility, (/Pa) Water compressibility, (/Pa) Capillary pressure, (Pa) Iteral Cotrol Valve Received : 4 th September 005 ; Accepted : 7 th August 006 REFERENCES [] Asheim, H., Maximizatio of Water Sweep Efficiecy by Cotrollig Productio ad Ijectio Rates, paper SPE 8365,(99). [] Brouwer, D.,R., Jase, J.,D., va der Starre, S., ad Va Kruijsdijk, C.P.J.W., Recovery Icrease through Water Floodig usig Smart Well Techology, paper SPE 68979, (000). [3] Brouwer, D., R., ad Jase, J.D., Dyamic Optimizatio of Water Floodig with Smart Wells Usig Optimal Cotrol, paper SPE 7878, (00). [4] Sudaryato, B., ad Yortsos, Y., C., Optimizatio of Fluid Frot Dyamics i Porous Media Usig Rate Cotrol,. I. Equal Mobility Fluids, Phys. of Fluids, (7), 656, (000). [5] Yete, B., Durlofsky, L.J. ad Aziz, K., Optimizatio of Smart Well Cotrol, paper SPE 7903, (00). [6] Box, G.E.P., Huter, W.G. ad Huter, J.S., Statistics for Experimeters, Joh Wiley & Sos, (978). [7] Friedma, F., Chawathe, A. ad Larue, D., Assessig Ucertaity i Chaelized Reservoirs usig Experimetal Desigs, paper SPE 76,(000). [8] Atkiso, A. C. ad Doev A. N., Optimum Experimetal Desigs, Oxford Uiversity Press, Oxford, UK, (99). [9] Cook, R.D. ad Nachtsheim, C.J., A Compariso of Algorithms for Costructig exact D-Optimal Desigs, Techometrics, (3), August (980). [0] Narayaa, K., Applicatios for Respose Surfaces 04
Ira. J. Chem. Chem. Eg. Sesitivity Aalysis of Water Vol. 6, No., 007 i Reservoir Egieerig, MS Thesis, Uiversity of Texas, Austi, (999). [] Draper, N. R. ad Li, D. K. J., Small Respose- Surface Desigs, Techometrics, 3,87 (990). [] Box, G.E.P. ad Draper, N.R., Empirical Model- Buildig ad Respose Surface, Joh Wiley ad Sos, (987). [3] Motgomery, D. C. ad Peck, E. A., Itroductio to Liear Regressio Aalysis, Joh Wiley ad Sos, (98). 05