SPE ,.(1) Introduction Several different smart well systems are available with different functionality. The simplest systems consist of sliding

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1 SPE Near-Well Reservir Mnitring Thrugh Ensemble Kalman Filter Geir Nævdal, Trnd Mannseth, SPE, and Erlend H. Vefring, SPE, RF- Rgaland Research Cpyright 2002, Sciety f Petrleum Engineers Inc. This paper was prepared fr presentatin at the SPE/DOE Imprved Oil Recvery Sympsium held in Tulsa, Oklahma U.S.A., April This paper was selected fr presentatin by an SPE Prgram Cmmittee fllwing review f infrmatin cntained in an abstract submitted by the authr(s). Cntents f the paper, as presented, have nt been reviewed by the Sciety f Petrleum Engineers and are subect t crrectin by the authr(s). The material, as presented, des nt necessarily reflect any psitin f the Sciety f Petrleum Engineers, its fficers, r members. Papers presented at SPE meetings are subect t publicatin review by Editrial Cmmittees f the Sciety f Petrleum Engineers. Electrnic reprductin, distributin, r strage f any part f this paper fr cmmercial purpses withut the written cnsent f the Sciety f Petrleum Engineers is prhibited. Permissin t reprduce in print is restricted t an abstract f nt mre than 300 wrds; illustratins may nt be cpied. The abstract must cntain cnspicuus acknwledgment f where and by whm the paper was presented. Write Librarian, SPE, P.O. Bx , Richardsn, TX , U.S.A., fax Abstract In the management f reservirs it is an imprtant issue t utilize the available data in rder t make accurate frecasts. In this paper a nvel apprach fr frequent updating f the near-well reservir mdel as new measurements becmes available is presented. The main fcus f this apprach is t have an updated mdel usable fr frecasting. These frecasts shuld have initial values that are cnsistent with recent measurements. The nvel apprach is based n utilizing a Kalman filter technique. The idea behind the Kalman filter is t incrprate the infrmatin frm the measurements int the current estimate f the state f the mdel, taking int accunt the uncertainty that belngs bth t the state f the mdel and the measurements. The uncertainty f the mdel is updated simultaneusly with the mdel itself. A benefit f this apprach cmpared t usual histry matching is that the initial values fr the frecasts will be in better agreement with the current measurements. Originally, the Kalman filter had shrtcmings fr large, nn-linear mdels. During the last decade, hwever, Kalman filter techniques has been further develped, and applied successfully fr such mdels within ceangraphic and hydrdynamic applicatin. This wrk is based n use f the ensemble Kalman filter. The ensemble Kalman filter is easy t implement, and have sme gd prperties fr nn-linear prblems. Here, we demnstrate the use f this technique within nearwell reservir mnitring, fcusing n its perfrmance in frecasting the future prductin. Intrductin Several different smart well systems are available with different functinality. The simplest systems cnsist f sliding sleeves which nly can be pen r shut and withut any mnitring. The mst advanced systems cnsist f infinitely variable chkes and extensive mnitring like pressure, temperature, multi-phase metering, and resistivity and seismic sensrs fr tracking near well fluid cntacts. The smart well systems are mtivated by the pssibility f imprved reservir management. Remte chking r shutting znes with pr perfrmance will cause an immediate respnse n the well perfrmance withut any expensive well interventin. Anther benefit f smart well systems is imprved reservir mnitring. Smart wells systems add value by enhancing wrkflw cycles cntaining the key elements f measurement, mdeling and cntrl. Several papers 1,2,3,4 have been presented where the pssible benefits f using smart well systems have been quantified. In all these papers the reservir mdel is assumed t be knwn. Hwever, a key element in the measurement, mdeling and cntrl lp is hw t update the near well reservir mdel based n the measurements. This is the fcus f the present paper. A nvel apprach fr updating a near-well reservir mdel based n measurements in the well will be presented. The apprach applies a Kalman filter technique and bth the reservir prperties and the state f the reservir is updated. Benefits f this apprach is that the initial values f the frecasts will be in better agreement with the current measurements and that the methdlgy is well suited fr frequent updating f the near well mdel. An alternative methdlgy fr updating the near-well reservir mdel cnsist in finding the reservir prperties which gives the least difference between measured data and mdel results within a given time interval. 5 We start by describing the reservir mdel. Then the ensemble Kalman filter methdlgy applied t near well reservir mdeling is presented. Examples f applicatin f the methdlgy are given, and finally sme cnclusins are drawn. The reservir mdel The mdel equatins fr tw-phase, immiscible prus-media flw with istrpic permeability are (see, e.g. Aziz & Settari 6 ) k µ r k B φs t B ( p γ z) = + q,.(1)

2 2 G. NÆVDAL, T. MANNSETH AND E. H. VEFRING SPE k µ S rg g + S g g k B g = 1 p p = P φs t B g ( pg γ g z) = + qg g,... (2),..(3) ( S g ) (4) We assume that the prsity is knwn all ver the reservir, while the permeability is unknwn, alng with the slutin variables; il pressure and gas saturatin. The remaining quantities in equatins (1) (4), except the external vlumetric flw rates, are knwn functins f the slutin variables. The external vlumetric flw rates are specified t be zer at uter bundaries. A multi-segment well mdel 7 is applied t cuple reservir and wellbre flws. The ensemble Kalman filter fr near-well reservir mnitring The values f several imprtant quantities, as fr instance prsity, permeability and relative permeability, in the reservir mdel varies in the reservir, and quantities as prsity and permeability are nt even accessible fr measurement except frm cre samples. The reservir flw depends t large extent n these quantities, and therefre the quality f the frecasts made frm reservir simulatins depends n the ability t give reasnable values t these quantities. The apprach suggested in this paper is t imprve the frecasts by updating the relevant physical quantities, based n the measurements that becme available during prductin, as well as using the infrmatin gained frm cre samples. A similar apprach, t the ne presented here, has been applied in the study f underbalanced drilling 8. The Kalman filter 9 was riginally develped fr linear mdels. An early apprach fr treating nn-linear mdels is the extended Kalman filter 10. A prblem with the extended Kalman filter is that the number f simulatins needed at each step is at the same rder as the number f state variables. Fr large systems this will generally be t time-cnsuming. The extended Kalman filter is based n linearizatin and has shrtcmings fr strngly nn-linear mdels. Recently, prgress has been made addressing bth these prblems, and new variants f Kalman filters have been applied within meterlgy and ceangraphy. One f these filters, the ensemble Kalman filter 11 is very easy t implement, and we have used this fr estimating ur unknwn mdel parameters. Using the Kalman filter it is pssible t cmbine the infrmatin btained frm the measurements with the mdel t get an imprved estimate f the state vectr f the system. Our state vectr cntains the values fr each grid blck f the time dependent variables pressure and gas saturatin as well as a value fr the permeability in each grid blck. Such an extensin f the state vectr t include mdel parameters has been applied bth in the study f underbalanced drilling 8 and in a test mdel used in atmspheric research 12. In these applicatins the number f mdel parameters is in the range frm 1-9, a mdest number cmpared t the number f mdel parameters included in the state space in ur applicatins. In additin t extend the state parameters with mdel parameters, it is necessary t include all measurements that is cnnected t the state variables by a nn-linear relatin in the state vectr, due t the frmulatin f the ensemble Kalman filter. This means that the state vectr has t be further augmented with the measurements btained frm pressure sensrs and multi-phase metering lcated in the well. We will nw give a shrt presentatin f the implemented filter, and discuss sme details n the actual implementatin fr this study. This presentatin fllws clsely the presentatin given in Lrentzen 8. T cmbine the infrmatin frm the measurements with the mdel in a prper way, we need bth t knw the uncertainty in the current estimate f the state and the uncertainty in the measurements. We assume that the errrs in the measurements are statistically independent, and with knwn variances. This gives a cvariance matrix Σ fr the measurement errrs. In the ensemble Kalman filter the cvariance matrix f the estimate f the states are btained using statistics built by an ensemble f state vectrs. After the ensemble is updated by taking int accunt new measurements we g thrugh the fllwing steps. Dente the state vectr fr the th member f the ensemble after inclusin f the measurement by. Each state a vectr is used as initial value t the simulatr fr a frward simulatin that is run t the time when the next measurements are taken int accunt. The th state vectr prir t the inclusin f the next measurements is f a s = f(s ) + ψ, (5) where s ) dentes the updating f the state vectr dne by f( a the simulatr and ψ is a stchastic cntributin representing the mdel errr. The mdel errr we use is nrmally distributed with zer mean and cvariance matrix Ψ. Mre details n the specificatin f Ψ will be given belw. T take int accunt the measurements we use the cvariance matrix f the ensemble arund the ensemble mean. The mean value f the ensemble is given by 1 n f ŝ = i = s 1,..(6) n and the ensemble cvariance matrix is 1 n n f f T R = (s ŝ)(si ŝ), (7) i= 1 = 1 n where n is the number f members in the ensemble. Fr prper use f the filter an ensemble f bservatins is needed 13. This is defined by d = d + ε, (8) where d is the actual bservatin and ε is drawn frm a nrmal distributin with zer mean and cvariance matrix Σ. s

3 SPE NEAR-WELL RESERVOIR MONITORING THROUGH ENSEMBLE KALMAN FILTER 3 The bservatin vectr d is related t the state vectr s thrugh the equatin d = Hs, fr an apprpriate matrix H. The state vectrs in the ensemble are updated using the gain matrix T T 1 G = RH (HRH + Σ),.(9) thrugh the equatin a f f s = s + G(d Hs ) (10) A mar issue with the ensemble Kalman filter is the size f the ensemble. The ptimal size f the ensemble fr ur applicatin is a subect fr further research. Experience in the ceangraphic science 14 has indicated that the filter may functin using a size f the ensemble in the range We have chsen t use 100 members in the ensemble. This means that 100 frward simulatins are needed. With increasing cmputatinal pwer, the challenge faced because f the size f the ensemble may be reduced, and it shuld be remarked that the ensemble Kalman filter is well suited fr parallelizatin. A mdificatin f the ensemble that reduces the cmputatinal burden has been prpsed 15, and it will be investigated in further wrk if this mdificatin is suitable fr the present prblem. In the specificatin f the cvariance matrix fr the mdeling errr,ψ, we make the assumptin that the dminating term is the uncertainty in the permeability. The cvariance matrix fr the mdel errr is blck-diagnal, with tw blcks. The first blck cnsists f the mdel errr in the lgarithm f the permeability. This is mdeled using a distance dependent crrelatin functin, such that the permeability in grid blcks that are lcated clse t each ther are updated in a crrelated manner, whereas there is small crrelatin in the updating f the permeability between grid blcks that are lcated far apart. Our assumptins is that the mar features f the mdel uncertainty is taken int accunt by the mdel uncertainty in the permeability, and therefre we nly use a very small, and uncrrelated, mdel errr fr the states representing the pressure and saturatin f the grid blcks as well as fr the state variables that are included t take int accunt the nnlinear well measurements. It is ur experience that prper specificatin f the cvariance matrix fr the mdeling errr is crucial t get gd perfrmance f the filter. Obtaining guidelines fr this specificatin will be an imprtant tpic fr further research. In the examples we present, all measurements are generated synthetically by running the mdel with a given permeability, and adding nise t the btained values t generate measurements. As cvariance matrix, Σ, fr the measurement errr we have used the same cvariance matrix as used when generating the measurements. In a field implementatin, the cvariance matrix fr the measurement errrs Σ shuld take int accunt the uncertainty in the measurement devices, but als include uncertainty in the psitining f the measurement gauges, and inaccuracies due t the applied numerical methd 16. Examples Cmmn setup f the examples. In the examples we present, we have kept the same reservir and well cnfiguratin. The scenari is ne hrizntal prducer, an immiscible tw-phase system with il and gas. The PVT prperties and relative permeability curves are adapted frm 17, and kept fixed bth while generating data and while running the ensemble Kalman filter. The reservir grid has dimensin , where the last crdinate refers t the vertical directin. The dimensin f each grid blck is ft. We have used a cnstant prsity in the reservir, except fr the tp layer which is used t emulate a gas cap. The prsity f the grid blcks in the tp layer is set t 1000, fr all ther grid blcks the prsity is 0.3. The depth f the tp layer is 6970 ft. The well has its heel in grid blck (4,1,14), and it is perfrated in three grid blcks, (5,1,14), (9,1,14) and (13,1,14) (see Figure 1). The depth f the well is 7375 ft, crrespnding t the middle f grid blck 14 in the vertical directin (where the vertical directin is numbered frm the tp). In the perfrated blcks, the annulus sectin is cnnected t the prductin tubing f the well thrugh chkes. While using the filter we assume that the fllwing quantities are available: Estimate f the permeability f the grid blcks penetrated by the well (frm cre samples), in additin t measurements during prductin. The quantities measured during prductins are the bttm-hle pressure and pressure in the three annulus sectins f the well, the ttal prductin f il and gas, and the inflw f these tw phases thrugh each annulus sectin. The uncertainties in the different measurements are shwn in Table 1. Table 1: Measurement uncertainties. Quantity Uncertainty (1 std. deviatin) Permeability f grid blck 3 % Pressure 4 psia Oil saturatin 10 % Gas saturatin 10 % Prductin data are sampled at every 0.1 days the first day, thereafter nce every day. The knwn permeability values are taken int accunt every time the state vectr is updated. The permeability used in the examples is shwn in Figure 1. The well is shwn in red (perfrated znes) and white. The heel f the well is lcated t the left. The well is set t prduce at an il rate f 1500 bbl/day, but with a limitatin n the gas prductin n Mscf/day. The difference between the tw examples is in the chke settings. In Example the pening f the chke in zne 3 is ne half f the pening used in zne 1 and 2. In example 3 the pening f the chkes is the same fr all three znes. In Example 1 the limitatin n gas prductin is met after 83 days, in Example 2 after 96 days. In this setting, the fcus has nly been n the ability t prduce reliable frecast fr the inflw, but cmbining

4 4 G. NÆVDAL, T. MANNSETH AND E. H. VEFRING SPE the technique fr prducing frecasts with steering is a challenging prblem. Result frm Example 1. Figures 2-8 shw the true slutin, the measurements and the estimated slutin fr different measured quantities. Figure 2 shws the bttm-hle pressure. The uncertainties in the pressure measurements are lw, and there is n visible difference between the estimated pressure and the true pressure. The same hlds fr the pressure measurements in each f the annulus sectins. Figures 3 5 shw the prductin f il fr the three inflw znes. One can bserve that the estimated rates fllw the true curves quite clsely, althugh the measurements are nisy. A similar bservatin can be dne fr the gas inflws, see Figure 6 8, althugh the estimate are fluctuating mre when the inflw is increasing. Fr the ttal prductin f il and gas the filter and the true curve cincides, s the plts are mitted. We evaluate the perfrmance f the ensemble Kalman filter by cnsidering the quality f the frecasts f prductin. Figures 9 14 shw frecasts f the il and gas inflw t each f the three inflw znes. The first frecast is dne after 0.1 days, which is the first measurement f the prductin data. At this pint the frecast f the gas inflw is misleading, it is predicted that the highest gas inflw will be in Zne 3, nt in Zne 1. While mre prductin data becmes available, this errr in the frecasts is gradually remved. The next frecast is shwn after 26 days. This is befre there is any significant gas inflw t the well. The quality f the frecasts in zne 1 and zne 3 are significantly imprved cmpared t the frecasts after 0.1 days. The last frecast is shwn after 73 days. There has been sme further imprvement in the frecast f the future gas prductin frm each f the znes. Result frm Example 2. Figures shw the true slutin, the synthetic measurements, and the estimated slutin fr the il and gas inflw t the three znes in Example 2. Figures shw the prductin f il fr the three inflw znes. One can bserve that the estimated rates fllw the true curves quite clsely, althugh the measurements are nisy. It seems that the filter wrks sme what better fr the znes with higher inflw rates. Figures shw the prductin f gas fr the three inflw znes. There is very lw difference between the estimated value and true value when the gas inflw is lw. As the inflw increases, the estimated curve has larger fluctuatins abut the true curve. There are tw factrs that bth are reasnable explanatins fr this behavir. One f the factrs is that the measurement nise is relative, and therefre larger fr larger inflws. In additin the system is run frm an equilibrium state, leading t an increased uncertainty abut the gas saturatin in the reservir with time. Figure shw frecasts f the il and gas inflw t the three znes. The frecasts are shwn after 0.1, 16 and 64 days. Althugh the limitatin n ttal prductin f gas is met later in this example than the previus ne, there is significant gas inflw earlier in zne 3 fr this example. The perfrmance f the frecasts f gas prductin after 64 days is much better than after nly 0.1 days. There are indicatins that the early measurements are mre imprtant fr imprving the frecast than later measurements, and the frecast f the gas prductin seems t be mre influenced by the updating than the il prductin in these tw examples. T study the effect f the measurements at different time intervals, and the pssibilities fr tuning different variables in the filter is a tpic fr further research. Cnclusins A new methdlgy using the ensemble Kalman filter technique fr frecasting prductin f a near-well reservir has been presented. The technique has been studied thrugh synthetic examples. With this technique, ne is able t track the prductin f the tw phases in each f the three inflw znes with much better accuracy than if these values are btained slely frm the measurements f the same quantities. Even mre imprtant is it that the reservir mdel is updated, such that frecasts can be cmputed which are cnsistent with the recent measurements. This culd be further explited in cntrl f smart wells. It is seen that, generally, the frecasts are imprved while mre measurements becmes available. Nmenclature B = Frmatin vlume factr d = Measurement G = Kalman gain H = Measurement matrix k = Permeability k r = Relative permeability n = Number f members in ensemble P = Capillary pressure p = Pressure q = External vlumetric flw R = Ensemble errr cvariance matrix S = Saturatin s = Member f ensemble t = Time z = Spatial vertical crdinate γ = Prduct f mass density and acceleratin f gravity µ = Viscsity Σ = Measurement errr cvariance matrix ε = Measurement errr Ψ = Mdel errr cvariance matrix ψ = Mdel errr Φ = Prsity Subscripts g = Gas = Related t the ensemble = Oil

5 SPE NEAR-WELL RESERVOIR MONITORING THROUGH ENSEMBLE KALMAN FILTER 5 Superscripts a = Analyzed. f = Frecast. Acknwledgements This wrk has been supprted financially by the Nrwegian Research Cuncil. References 1. Jalai, Y., Bussear, T. and Sharma, S., Intelligent Cmpletin Systems The Reservir Ratinale, paper SPE presented at the 1998 SPE EurpeanPetrleum Cnference held in the Hague,The Netherlands, Octber Sinha, S., Kumar, R.,Vega, L. and Jalai, Y., Flw Equilibratin Twards Hrizntal Wells Using Dwnhle Valves, paper SPE68635 presented at the SPE Asia Pacific Oil and Gas Cnference and Exhibitin held in Jakarta, Indnesia, April Yeten, B. and Jalali Y., Effectiveness f Intelligent Cmpletins in a Multiwell Develpment Cntext, paper SPE presented at the 2001 SPE Middle East Oil Shw held in Bahrain, March Yu, S. and Davies, D. R., The Mdelling f Advanced Intelligent Well An Applicatin, paper SPE presented at the 2000 SPE Annual Technical Cnference and Exhibitin, Dallas, Texas, 1 4 Octber Mannseth, T., Nrdtvedt, J. E., and Nævdal, G.: Optimal Management f Advanced Wells thrugh Fast Updates f the Near-Well Reservir Mdel, Prceedings f the 7 th Eurpean Cnference n the Mathematics f Oil Recvery, Baven, Italy, 5 8 Spetember, Aziz, K. and Settarri, A.: Petrleum Reservir Simulatin. Applied Science, Lndn, (1979). 7. Eclipse Technical Descriptin. 2000A. Schlumberger. (2000). 8. Lrentzen, R.J., Felde, K.K., Frøyen, J., Lage, A.C.V.M., Nævdal, G., and Vefring, E.H.: Uncerbalanced and Lwhead Drilling Operatins: Real Time Interpretatin f Measured Data and Operatinal Supprt. Paper SPE presented at the 2001 SPE Annual Technical Cnference and Exhibitin held in New Orleans, Lusiana, 30 September 3 Octber (2001). 9. Grewal, M.S. and Andrews, A.P.: Kalman Filtering:Thery and Practice. Prentice Hall, Englewd Cliffs, New Jersey (1983). 10. Maybeck, P.S.: Stchastic Mdels, Estimatin, and Cntrl. Vl. 2. Academic Press, New Yrk (1982). 11. Evensen, G.: Sequential Data Assimilatin with Nnlinear Quasi-gestrphic Mdel using Mnte Carl Methds t frecast Errr Statistics. J. Gephys. Res. Vl. 99 (C5), pp , (1994). 12. Andersn, J. L.: An Ensemble Adustment Kalman Filter fr Data Assimilatin. Mnthly Weather Review. Vl. 129, pp , (2001). 13. Burgers, G., van Leeuvwen, P. J., and Evensen, G.: Analysis Scheme in the Ensemble Kalman Filter. Mnthly Weather Review. Vl. 126, pp , (1998). 14. Evensen, G.: Applicatin f Ensemble Integratins fr Predictability Studies and Data Assimilatin. Published in: Mnte Carl Simulatins in Oceangraphy, Prceedings 'Aha Hulik'a Hawaiian Winter Wrkshp, University f Hawaii at Mana, January , (1997). 15. Heemink, A.W., and Verlaan, M., and Seegers, A.J.: Variance reduced Ensemble Kalman Filtering. Mnthly Weather Review. Vl. 129, pp , (2001). 16. Chn, S.E.: An Intrductin t Estimatin Thery. Jurnal f the Meterlgical Sciety f Japan. Vl. 75, pp , (1997). 17. Eclipse Reference Manual 2000A. Schlumberger (2000). Figure 1. The true permeability. The well is shwn in red and white. The perfrated part f the well is red. The permeability is given in md. Figure 2. Bttm hle pressure in Example 1. Figure 3. Oil inflw in zne 1 (grid blck (5,14), heel f well) in Example 1.

6 6 G. NÆVDAL, T. MANNSETH AND E. H. VEFRING SPE Figure 4. Oil inflw in zne 2 (grid blck (9,14)) in Example 1. Figure 7. Gas inflw in zne 2 in Example 1. Figure 5. Oil inflw in zne 3 (grid blck (13,14), te f well) in Example 1. Figure 8. Gas inflw in zne 3 in Example 1. Figure 6. Gas inflw in zne 1 in Example 1. Figure 9. Frecast f il inflw in zne 1 in Example 1.

7 SPE NEAR-WELL RESERVOIR MONITORING THROUGH ENSEMBLE KALMAN FILTER 7 Figure 10. Frecast f il inflw in zne 2 in Example 1. Figure 13. Frecast f gas inflw in zne 2 in Example 1. Figure 11. Frecast f il inflw in zne 3 in Example 1. Figure 14. Frecast f gas inflw in zne 3 in Example 1. Figure 12. Frecast f gas inflw in zne 1 in Example 1. Figure 15. Oil inflw in zne 1 in Example 2.

8 8 G. NÆVDAL, T. MANNSETH AND E. H. VEFRING SPE Figure 16. Oil inflw in zne 2 in Example 2. Figure 19. Gas inflw in zne 2 in Example 2. Figure 17. Oil inflw in zne 3 in Example 2. Figure 20. Gas inflw in zne 3 in Example 2. Figure 18. Gas inflw in zne 1 in Example 2. Figure 21. Frecast f il inflw in zne 1 in Example 2.

9 SPE NEAR-WELL RESERVOIR MONITORING THROUGH ENSEMBLE KALMAN FILTER 9 Figure 22. Frecast f il inflw in zne 2 in Example 2. Figure 25. Frecast f gas inflw in zne 2 in Example 2. Figure 23. Frecast f il inflw in zne 3 in Example 2. Figure 26. Frecast f gas inflw in zne 3 in Example 2. Figure 24. Frecast f gas inflw in zne 1 in Example 2.