Reservoir Multiscale Data Assimilation Using the Ensemble Kalman Filter

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1 Applied Mathematis doi:0436/am009 Published Online February 0 ( Reservoir Multisale Data Assimilation Using the Ensemble Kalman Filter Abstrat Santha R Akella Department o Earth and Planetary Sienes Johns Hopkins University Baltimore USA santhaakella@jhuedu Reeived June 00; revised November 00; aepted November 5 00 In this paper we propose a way to integrate data at dierent spatial sales using the ensemble Kalman ilter (EnKF) suh that the inest sale data is sequentially estimated subjet to the available data at the oarse sale (s) as an additional onstraint Relationship between various sales has been modeled via upsaling tehniques The proposed oarse-sale EnKF algorithm is reursive and easily implementable Our numerial results with the oarse-sale data provide improved ine-sale ield estimates when ompared to the results with regular EnKF (whih did not inorporate the oarse-sale data) We also tested our algorithm with various preisions o the oarse-sale data to aount or the inexat relationship between the ine and oarse sale data As expeted the results show that higher preision in the oarse-sale data yielded improved estimates Keywords: Kalman Filter Reservoir Engineering Unertainty Quantiiation Multisale Data Introdution The prinipal objetive o data assimilation methods [] is to ombine the inormation provided by measured data and a (numerial) oreast model to obtain an improved estimate o the system state (and parameters) Unlike variational methods whih require availability o omplex adjoint models or data assimilation the ensemble Kalman ilter (EnKF) an be quikly implemented and one an also obtain unertainty estimates via error varianeovariane propagation; see [] and reerenes therein or urther details The EnKF is a sequential Monte Carlo method based on Bayes theorem The method is inreasingly being used or estimating unknown model state and parameters in various geologial and hydrologial models [3] One o the major problems in subsurae haraterization is the huge unertainty in the knowledge o hydroarbon reservoir permeability and porosity Sine the low o hydroarbons suh as oil and gas through the subsurae ormation ritially depends on the geologial rok properties it is important to aurately know these properties This artile ouses on methods to obtain more aurate quantiiation o the reservoir rok properties using measured data Broadly speaking the measured data used or desription o reservoir porosity and permeability haraterization onsist o stati and dynami data Stati data suh as well logs ore samples an resolve heterogeneity at a sale o a ew inhes or eet with high reliability However dynami data suh as rational low (deined as the ratio o the injetion luid to the total luid produed at the prodution wells; or water ut) pressure transient and traer test data typially san the length sales omparable to the inter-well distanes Additional dynami data suh as time-lapse seismi images [4] an provide improved spatial sampling but at a lower preision A majority o previous studies on unertainty quantiiation in reservoir perormane oreasting using EnKF have mostly dealt with integration o dynami data (or eg [5-7]) However it is widely reognized that integration o additional multisale data ould urther redue the unertainty (see [89] and reerenes therein) With that perspetive integration o data at oarse-and ine-sales is an important objetive and is addressed in this paper We use the EnKF to estimate ine-sale ields or subsurae haraterization Also our method ould be generalized to other sequential data assimilation methods suh as partile iltering (where rather than updating the ensemble members model state we update the probability assigned to eah ensemble member based on model data misit) The main reason why we used EnKF in this paper is beause it requires

2 66 ewer ensemble members than the partile ilters see [3] and reerenes therein or urther details In this paper apart rom the water ut data we onsider oarse-sale measured data as well The oarse-sale data is assumed to be permeability at some speiied level o preision The unknown variables: permeability at the ine-sale are estimated using a modiiation to the EnKF algorithm linking the data at dierent sales via upsaling It is important to resolve ine-sale heterogeneity or various purposes suh as enhaned oil reovery environmental remediation et The main idea behind upsaling is to obtain an eetive oarse-sale permeability whih yields the same average response as that o the underlying ine-sale ield loally Single phase low upsaling proedures or two phase low problem have been disussed by many authors; see eg [0] and also Setion 3 We will reer to our proposed variant o EnKF as oarse-sale EnKF Assimilation using dynami data suh as rational low data only is thereore reerred to as regular EnKF The oarse-sale permeability data ould be obtained either rom geologi onsideration or oarse-sale inversion o dynami rational low data on a oarse grid as onsidered in [8] This oarse-sale stati data an be viewed as a onstraint whih is to be satisied up to the presribed variane or obtaining the ine-sale estimates in every data assimilation yle Upsaling methods relate the solution at the inesale to the oarse-sale thereore in the Kalman ilter ontext it amounts to modeling a nonlinear observation operator In our oarse-sale EnKF approah we use the measured data in bathes suh that the estimate with one data beomes a prior while assimilating the other observation (see Setion 3 or urther details) Though in this paper we used oarse-sale data at only one sale our approah an be easily generalized to assimilate data at multiple sales by appropriately modeling the linkage between dierent sales Also our upsaling method is independent o the underlying inesale ield For the purpose o sel-ontendness and notational larity we briely review the governing equations sequential data assimilation using the ensemble Kalman ilter in Setion whih is ollowed by a desription o the oarse-sale EnKF algorithm (Setion 3) For our numerial results (Setion 4) we onsider a ive-spot pattern with the injetion well plaed in the middle o a retangular domain and our prodution wells loated at the verties o the retangle A reerene ase is used to provide true data whih is randomly perturbed to obtain syntheti measurements A omparison o the regular EnKF with the oarse-sale EnKF (Setions 4 and 4 respetively) shows that using oarse-sale permeability data (via oarse-sale EnKF) signiiantly improves the ine-sale estimates as well as uture rational low predition Bakground Fine-Sale Model In this paper we onsider two-phase low in a subsurae ormation under the assumption that the displaement is dominated by visous eets For simpliity we neglet the eets o gravity ompressibility and apillary pressure although our proposed approah is independent o the hoie o physial mehanisms Also porosity will be onsidered to be onstant The two phases will be reerred to as water and oil (or a non-aqueous phase liquid) designated by subsripts w and o respetively We write Dary s law or eah phase as ollows: krj S νj = κpr; S κ pr= h () j krw S kro S krw S w S= S= w o krw S w kro S o S ν = νw νo = S κ pr; ν S t () The above desriptions are reerred to as the ine-sale model o the two-phase low problem Here κ is the (ine-sale) permeability o the medium S is the total mobility j denotes phase visosity pr is the pressure h is the soure term and S denote porosity and water saturation (volume ration) respetively Sequential Estimation using EnKF Using dynami measured data suh as water ut we an sequentially estimate the unknown parameters (permeability porosity et) and state variables suh as pressure water saturation (two-phase low) and prodution data at well loations using the EnKF as disussed in [573] Following these previous works in this paper we assume that the only dynami data available is water ut data and that porosity is known The ombined state-parameter T to be estimated are given by Ψ = lnκ pr S W Where ln is natural logarithm o permeability ield and W denotes water ut; in order to distinguish observed water ut rom model predited water ut now o onwards we will denote the observed water ut data W by y The EnKF introdued in [4] is a sequential Monte Carlo method where an ensemble o model states evolve in state-spae with mean as the best estimate and spread o the ensemble as the error ovariane as summarized in the ollowing steps Eah o the ensemble members is

3 67 oreasted independently (in this work we negleted modeling errors) i n Ψ = F Ψ (3) where F is the oreast operator (Equations () ()) supersript i denotes the i th ensemble member; now onwards we will drop the time subsript The ensemble mean and ovariane are deined as N e e i= i n i Ψ = N Ψ (4) T P A' A ' (5) N e N where A '= b b b e i i b = Ψ Ψ and N e is the number o ensemble members The observation vetor or eah ensemble member is given by i y t = H Ψ ν (6) t where H Ψ is the observed data rom the truth and i ν represents observational errors whih are iid samples [5] rom a normal distribution with zero mean and variane R We note that i only the water ut data is being measured the mapping rom model-to-observational spae H is trivially equal to 000I sine T Ψ = lnκ pr S W The oreasted ensemble (Equation (3)) is updated by assimilating the observed data Ψ i i i i Ψ K y H Ψ (7) i where K is the Kalman gain given by K T T = P H HP H R Computationally eiient implementation o the EnKF is disussed or eg in [6] We use the above set o orreted ensemble states N i e Ψ in the simulation 0 model (Equation (3)) to predit until the next set o observational data is available i 3 Coarse-Sale Constrained EnKF The EnKF presented so ar used only the dynami prodution data (water ut) y with error ν = y H t Ψ ν 0R to update the ensemble (Equation (7)) In addition to y i we are also given stati data (as mentioned in the Introdution) whih is another set o independently measured data z Assuming that the orresponding measurement error is given by t ω = z U Ψ ω 0Q ; U : Ψ z Its likelihood is given by p exp (8) T z Ψ zuψ Q zuψ z I this stati data z orresponds to oarse-sale permeability data [8] then U = 000 Where : κ κ is a nonlinear mapping that maps the ine-sale permeability ield ( κ ) to oarse-sale ield ( κ ) via an upsaling proedure (eg [78]) details are provided in Setion 3 Note that by deinition the errors in water-ut data y and oarse-sale permeability κ data don t inluene eah other sine they are entirely dierently measured quantities Now our goal is to obtain an estimate whih is based on both o the above dynami and stati data The likelihood o y is given by T p y Ψ exp yh Ψ R yhψ y The probability distribution untion (pd) o the predited ensemble T p Ψexp ΨΨ P ΨΨ where Ψ and P are the predited ensemble mean and ovariane respetively (Equations (4) and (5)) Then using Bayes theorem we obtain p Ψ zy p Ψ zy = p zy pzyψ pψ = pzyψ pψ p zy = p z Ψ p y Ψ p Ψ pψ y The last term in above equation implies that the two independent data y and z an be sequentially assimilated in the ollowing two steps We irst assimilate observation y to obtain an intermediate ensemble N i e Ψ as 0 disussed in Setion p Ψ p Ψ y exp (9) y This intermediate ensemble and likelihood in Equation (8) an then be ombined to obtain the inal estimate N i e i0 Ψ p Ψ pψ zy exp y z (0) Thereore in a least-squared sense the inal estimate p Ψ zy whih orres- maximizes the posterior pd i

4 68 ponds to the minimum o = z y See Appendix A or urther details (where we show that the solution Ψ i orresponds to the minimum o or th any i ensemble member) The oarse-sale EnKF algorithm is detailed in Appendix B 3 Upsaling Methods In brie the main idea behind upsaling o absolute ine-sale permeability is to obtain eetive oarse-sale permeability or eah oarse-grid blok One the upsaled absolute permeability is omputed the original equations are solved on the oarse-grid without hanging the orm o relative permeability urves This is an inexpensive alulation sine the pressure update involves only solving the pressure equation on the oarse-grid and one an take larger time step or solving the transport equation In our numerial simulations the ine-grid is oarsened 0 times in eah diretion These kinds o upsaling tehniques in onjuntion with the upsaling o absolute permeability have been used in groundwater appliations (see eg [8]) The link between the oarse and the ine-sale permeability ields is usually nontrivial beause one needs to take into aount the eets o all the sales present at the ine level In the past simple arithmeti harmoni or power averages have been used to link properties at various sales These averages an be reasonable or low heterogeneities or or volumetri properties suh as poros- ity For permeabilities simple averaging an lead to inaurate and misleading results In this paper we use the low-based upsaling methods using loal solutions o the equations [79] First we briely desribe low based upsaling methods Consider the ine-sale permeability that is deined in the domain with underlying ine grid as shown in Figure On the same graph we illustrate a oarsesale partition o the domain To alulate the oarsesale permeability ield at this level we need to determine it or eah oarse blok The oarse blok permeability an be deined both using the solutions o loal or global problems The main idea used to alulate the oarse-sale permeability is that it should deliver the same average response as that o the underlying inesale problem loally The alulation o the oarsesale permeability based on loal solutions is shematially depited in Figure For eah oarse domain we solve the loal problems κ x () j =0 with some oarse-sale boundary onditions One o suh boundary onditions is given by j = and j =0 on the opposite sides along the diretion e j and no low boundary onditions on all other sides alternatively j = x j on For these boundary onditions the oarse-sale permeability is given by κ e e = κ x e dx () j l j l 0 div κ x 0 Figure Shemati illustration o upsaling (not to sale): bold lines indiate a oarse-sale partitioning while thin lines show a ine-sale partitioning within oarse-grid ells In this paper we upsaled a ine-grid to a 5 5 oarse-grid

5 69 where j is the solution o Equation () with presribed boundary onditions Various boundary ondition an have some inluene on the auray o the alulations inluding periodi Dirihlet et These issues have been disussed or eg in [9] In partiular or determining the oarse-sale permeability ield one an hoose loal domains that are larger than target oarse blok or Equation () Further Equation () is used in the domain where j are omputed in the larger domains with orret saling (see [9]) This way one redues the eets o the artiiial boundary onditions imposed on (or details see [9]) The use o the loal solutions Equation () or determining the permeability ield at dierent sales gives non-expliit relation or onditional distribution We denote by the loal operator that maps the loal ine-sale permeability ield κ into κ deined as above For our omputations we assume κ = κ (3) where are some random lutuations that represent inauraies in the oarse-sale permeability In reality sine we do not have the ine-sale ield κ available it is diiult to haraterize the exat (nature o the) error in upsaling However one o the soures o these lutuations are the errors assoiated with solving inverse problems on the oarse grid The other soure o the inauraies o measured oarse-sale permeability is due to the at that the inversion on the oarse grid does not take into aount the adequate orm o the oarse-sale models Here we assumed these errors to be normally distributed (urther details ollow in Setion 4) 4 Numerial Results For our numerial tests with the oarse-sale EnKF algorithm we use a ine grid (dimensionless domain size ) We onsider the oarse-sale permeability whih ould be obtained by oarse-sale inversion o rational low data on a oarse grid [0] This oarse-sale ield ould be thought as stati data whih is to be honored as onstraint (up to the data variane) in Equation (8) hene we need to always assimilate it in our oarse-sale EnKF algorithm An initial ensemble with dierent permeability realizations was generated using the sequential Gaussian simulation (SGSIM) [] We speiied a Gaussian variogram model with a orrelation length o 0 gridbloks in the x-diretion and 5 gridbloks in the y-diretion; one o For reservoir simulation appliations the SGSIM has been used [ 3] or generating initial ensemble members This approah yields independent and identially distributed multivariate normal random ields (onditioned to well log data) the realizations is used as the reerene ield (depited in Figure ) The rational low will be alulated based on the ine-sale model in Setion Porosity ( ) is assumed to be equal to 05 or all grid bloks For simpliity relative permeabilities k rj are assumed to be linear untions o water saturation (S): krw S = S kro S = S One injetion well at the enter o the ield (injetion rate: 74 m 3 /day) and our produing wells at the our orners (all with equal rate o 785 m 3 /day) were onsidered The model equations are solved with no low boundary onditions zero initial water saturation and disretizing the transport equation using irst order upwind inite volume method In Figure 3 we provide the predited rational low or 56 initial ensemble members along with the true rational low (obtained rom true permeability ield) To ompare our proposed oarse-sale onstrained EnKF results with the regular EnKF we will use the ollowing mean L -norm error Sine we know the true (ine and oarse-sale) ield or our syntheti problem true ie the true permeability ield denoting it by κ the error or any ensemble member is i i true e = κ κ i = N e Consider the L norm o the error or eah member i i j j e = e using whih we deine the mean L error as Ne i e = N e (4) i= e so that e gives us an indiation o the distane o entire true ensemble rom the true solution κ Sine ater every Figure Natural logarithm o true permeability ield

6 70 Figure 3 Frational low predition with 56 initial ensemble members (no data assimilation); ensemble members (green dots) ensemble mean (blue rosses) ompared with true water ut data (red open irles) observation we have updated ensemble members thereore we an monitor the variation o e over the time o assimilation; the suess o assimilation an thereore be related to the derease in e 4 EnKF with Frational Flow Data Only We start with a presentation o results with regular EnKF assimilating only water ut data Next we will disuss results with the oarse-sale EnKF The water ut data rom the reerene ield is assumed to be available every 00 days with mean zero and standard deviation o 00 (thereore R / =00 I 4 where I 4 is unit matrix o size 4 4 sine there are our produing wells) The observed data is assumed to be available up to 400 days hene we will perorm assimilation between 00 and 400 days A predition beyond interval o data assimilation up to 4000 days is also provided We seleted an ensemble o size 56 or presenting our data assimilation results We assimilated the above desribed measured data and using the assimilated permeability ield in Figure 4 we plot the assimilated water ut data along with the true data Comparing with the initial oreast in Figure 3 we observe that the assimilated ensemble better envelopes the true data We ompare the initial permeability ield beore assimilation (Figure 5(a)-(d)) or a ew ensemble members with the true ield in Figure and with those obtained ater assimilation in Figure 6(a)-(d); note that the entral South East-North West hannel is prominent but the eatures at the South West and North East orners are not well aptured Thereore assimilation o only water ut data helps in identiying only some o the important eatures 4 Coarse-Sale Constrained EnKF with Frational Flow and Coarse-Sale Permeability Data In addition to water ut prodution data the oarse-sale

7 7 Figure 4 Water ut predition using assimilated (regular EnKF or 400 days) ensemble members note the improved it o ensemble when ompared to that in Figure 3 permeability data as desribed in Setion 3 has been used as additional measured data Flow-based upsaling o reerene permeability ield is used as a proxy or inverted oarse ield Following our previous notation this oarse-sale permeability data will be denoted by z (Equation (8)) The mapping between state variables (at ine-sale) and observations (at oarse-sale) is given by U = 000 denotes low-based upsaling Exatly as in the previous setion we presribed the same requeny (o availability) and preision R or the rational low data Sine we use oarse-sale permeability as additional data it is to be assimilated whenever we assimilate water ut data A 5 5 oarse-sale data with mean zero and variane Q = qi 5 (we will present results with q =405 and 0) so that we an onsider the impat o oarse-sale data preision In Figure 7 we plot the variation o mean L error e (Equation (4)) with observation time at the oarse-sale or dierent values o q Figures 8(a) and (b) depit the orrelation between oarse-sale ensemble mean and true ields or q = 4 and 0 respetively As the preision o oarse-sale data is inreased ie or smaller variane we observe a larger derease in oarse-sale mean L error and higher orrelation with true oarse-sale ield (orrelation oeiient or q =40 respetively are ) beause smaller variane Q implies more striter oarse-sale data onstraint in Equation (8) Figure 9(a)-(d) depit the rational low using the inal permeability ield ater assimilation or dierent oarse-sale data preisions Figures 7 and 8(a)-(b) show that the oarse-sale data is being more aurately assimilated as it is made more preise Also notie the improved it o ensemble predition to the true data or more preise oarse-sale data; also when ompared to the regular EnKF results in Figure 4 Now we disuss the results regarding ine-sale ield In Figure 0 we plot the ine-sale mean L error or dierent values o q ; the oarse-sale EnKF yields muh lesser error than regular EnKF whih assimilated only rational low data The orrelation oeiient

8 7 (a) (b) () (d) Figure 5 Log permeabilities o a ew i-th initial ensemble members (beore data assimilation); let-right (a) i = 50 (b) 00 () 50 (d) 00 (a) (b) () (d) Figure 6 Same as above but ater assimilating water ut data with regular EnKF

9 73 Figure 7 Derease in e omputed at oarse-sale as data (rational low and 5 5 oarse-sale permeability data at variane Q= I q ) is assimilated using the oarse-sale EnKF algorithm Figure 8 Correlation between oarse-sale ensemble mean and true permeability ater assimilation or low and high preision in oarse data; (a) and (b): q = 4 0

10 74 (a) (b)

11 75 () (d) Figure 9 Same as in Figure 4 but using oarse-sale EnKF or data assimilation; lokwise (a)-(d): q = 4 0

12 76 Figure 0 Same as in Figure 7 but at ine-sale also shown is the error obtained with assimilation o rational low data only between ine-sale ensemble mean and true ields ater assimilating using regular EnKF is equal to 0409 while with the oarse-sale EnKF or q =40 in that order were and 066; note higher orrelation with the oarse-sale EnKF We observe that higher preision ie lower q does not neessarily imply least e or highest orrelation sine highly preise oarse-sale data is relatively more weighted than the rational low data Optimal value or the oarse-sale data variane an be obtained by prior alulation whih will be addressed in a uture study The inal permeability ield or a ew ensemble members ater assimilating with oarse-sale EnKF or q = is shown in Figure (a)-(d); all shown samples seem to be more loser to the true ield (Figure ) than those obtained with regular EnKF (Figure 6(a)-(d)) In partiular note that the low permeability region at the North East and high permeability at the South West orners are well aptured 5 Conlusions The EnKF is inreasingly being used or subsurae haraterization in various geologial and groundwater appliations to identiy ine-sale state and parameters So ar various implementations have been based on using dynami prodution data suh as water ut well pressures et or sequential data assimilation Only reently dynami data other than prodution data has been onsidered in the EnKF ontext ([34]) nevertheless the observed data to be assimilated was assumed to be at the inest sale For a number o reasons it is widely reognized that usage o additional multisale data ould urther redue the unertainty at the ine sale This is urther motivated by the inreasing popularity o oarsesale modeling In this light here we proposed assimilation o oarse-sale data along with water ut prodution data using oarse-sale EnKF The modiiation to the regular EnKF (assimilation o only water ut data) is ompletely reursive and easily implementable The relation between ine and oarse sales has been modeled via physis based upsaling whih ould be thought o as a nonlinear observation operator linking the oarsesale data to the unknown ine-sale variables In addition the proposed methodology ould be used in any other sequential data assimilation method as well and also

13 77 (a) (b) () Figure Same as in Figure 6 but assimilated using oarse-sale EnKF with q= or the variane o oarse-sale permeability data (d) also with any other upsaling method The oarse-sale EnKF was tested and ompared with the regular EnKF or a D syntheti heterogenuous true ield We onsidered oarse-sale permeability data as additional data on a 5 5 oarse grid This oarse-sale data was always assimilated along with water ut data The data variane was varied rom low to high to study its impat on assimilated results In all ases we observed that the assimilated ensemble mean oarse-sale ield or all varianes was highly orrelated to the true oarse-sale ield In addition lower variane in the oarse-sale data yielded higher orrelation The water ut data was better honored both or higher preision o oarse data and when ompared with regular EnKF As or the ine-sale permeability ield the oarse-sale EnKF yielded lesser error in an averaged L norm error taken wrt the reerene ield In addition a ew individual samples were piked to ompare the assimilated ields with dierent EnKF proedures; experiment with oarse-sale permeability data provided inal samples whih aptured most losely the eatures in the reerene ine-sale ield Though in our urrent paper we used only one oarsesale the proposed method an be easily implemented to integrate as many sales as required by the available data and is independent o the underlying ine-sale ield Based on our results we onlude that there ould be a singiniiant improvement in subsurae haraterization i aurate and additional data at oarse-sales is available In uture we plan to study the nature o errors involved in oarse-sale data and upsaling and in-turn their inluene on the proposed oarse-sale EnKF 6 Reerenes [] J M Lewis S Lakshmivarahan and S Dhall Dynami Data Assimilation: A Least Squares Approah (Enylopedia o Mathematis and its Appliations) Cambridge University Press Cambridge 006 doi:007/cbo [] G Evensen Data Assimilation: The Ensemble Kalman Filter Springer Berlin 006 [3] Y Liu and H V Gupta Unertainty in Hydrologi Modeling: Toward and Integrated Data Assimilation Framework Water Resoures Researh Vol W0740 doi:009/006wr [4] D E Lumley Time-Lapse Seismi Reservoir Monitoring Geophysis Vol 66 No 00 pp doi:090/4449 [5] G Nævdal L Johnson S Aanonsen and E Vering Reservoir Monitoring and Continuous Model Updating

14 78 Using Ensemble Kalman Filter SPE Journal Vol 0 No 005 pp [6] X-H Wen and W H Chen Real-Time Reservoir Model Updating Using Ensemble Kalman Filter Proeedings o SPE Reservoir Simulation Symposium 005 SPE January- February 005 Texas [7] Y Q Gu and D S Oliver The Ensemble Kalman Filter or Continuous Updating o Reservoir Simulation Models Journal o Energy Resoures Tehnology-Transations o the ASME Vol pp doi:05/34735 [8] S H Lee A Malallah A Datta-Gupta and D Higdon Multisale Data Integration Using Markov Random Fields SPE Reservoir Evaluation Engineering Vol 5 00 pp doi:08/76905-pa [9] Y Eendiev A Datta-Gupta V Ginting X Ma and B Mallik An Eiient Two-Stage Markov Chain Monte Carlo Method or Dynami Data Integration Water Resoures Researh Vol doi:009/004wr [0] M Christie Upsaling or Reservoir Simulation Journal o Petroleum Tehnology Vol 48 No 996 pp doi:08/3734-ms [] J W Barker and S Thibeau A Critial Review o the Use o Pseudo-Relative Permeabilities or Upsaling SPE Reservoir Engineering Vol 997 pp38-43 doi:08/3549-pa [] Y Eendiev A Datta-Gupta I Osako and B Mallik Multisale Data Integration Using Coarse-Sale Models Advanes in Water Resoures Vol 8 No pp doi:006/jadvwatres [3] D Devegowda E Arroyo-Negrete and A Datta-Gupta Eiient and Robust Reservoir Model Updating Using Ensemble Kalman Filter with Sensitivity-Based Covariane Loalization Proeedings o SPE Reservoir Simulation Symposium Houston February 007 doi:08/0644-ms [4] G Evensen Sequential Data Assimilation with a Nonlinear Quasi-Geostrophi Model Using Monte Carlo Methods to Foreast Error Statistis Journal o Geophysial Researh Vol pp 43-6 doi:009/94jc0057 [5] G Burgers P J van Leeuwen and G Evensen Analysis Sheme in the Ensemble Kalman Filter Monthly Weather Review Vol 6 No pp doi:075/ (998)6<79:asitek>0c O; [6] Y Chen and D Zhang Data Assimilation or Transient Flow in Geologi Formations Via Ensemble Kalman Filter Advanes in Water Resoures Vol pp 07- doi:006/jadvwatres [7] L J Durlosky Numerial Calulation o Equivalent Grid Blok Permeability Tensors or Heterogeneous Porous Media Water Resoures Researh Vol 7 No 5 99 pp doi:009/9wr0007 [8] L J Durlosky Coarse Sale Models o Two Phase Flow in Heterogeneous Reservoirs: Volume Averaged Equations and Their Relationship to the Existing Upsaling Tehniques Computational Geosienes Vol 998 pp 73-9 doi:003/a: [9] X H Wu Y R Eendiev and T Y Hou Analysis o Upsaling Absolute Permeability Disrete and Continuous Dynamial Systems Series B Vol 00 pp doi:03934/ddsb0085 [0] S Yoon A H Malallah A Datta-Gupta D W Vaso and R A Behrens A Multisale Approah to Prodution-Data Integration Using Streamline Models SPE Journal Vol 6 No 00 pp 8-9 [] Y Gu and D S Oliver History Mathing o the PUNQ-S3 Reservoir Model Using the Ensemble Kalman Filter SPE Journal Vol 0 No 005 pp 7-4 doi:08/8994-pa [] C V Deutsh Journel A GSLIB: Geostatistial Sotware Library and User s Guide Oxord Univiversity Press Oxord 99 [3] Y Dong Y Gu and D S Oliver Sequential Assimilation o 4D Seismi Data or Reservoir Desription Using the Ensemble Kalman Filter Journal o Petroleum Siene and Engineering Vol pp doi:006/jpetrol [4] J-A Skjervheim G Evensen S Aanonsen B Ruud and T Johansen Inorporating 4D Seismi Data in Reservoir Simulation Models Using Ensemble Kalman Filter SPE Journal Vol No pp 8-9 doi:08/95789-pa

15 Appendix A: Two Step Coarse-Sale Constrained Kalman Filter Estimate From Setion 3 T = ΨΨ P ΨΨ and T y = yh Ψ R yhψ For notational simpliity we will denote μ Ψ as μ and denote P by B Step (minimize y ): First we minimize the sum = y The gradient o above quadrati ost untional with respet to (wrt) Ψ is given by T = Ψ B Ψ μ H R yh Ψ Then the minimizer μ o satisies (we assume H to be linear) T B μ μ H R yhμ =0 Rearranging the above equation we get T T B H R H μ = B μ H R y (5) Note that the Hessian o wrt Ψ is given by T B H R H and or linear quadrati ost untionals the Hessian inverse is equal to the error ovariane matrix Thereore the error ovariane matrix B or μ is given by B T B H R H (6) Step (minimize g z ): We use μ B in T Ψ μ B Ψ μ g T = zuψ Q zuψ z Thereore the minimum μ o g z satisies T T ˆ B U Q U μ B μ U Q z Using Equations (6) and (5) we an rewrite above as T T B H R HU Q Uμ ˆ B T T B μ H R yu Q z r h s o eqation (4) We note in passing that B and R are ovariane matries and are positive deinite by onstrution and hene or our derivation purposes are ormally invertible It is trivial to show that ˆμ also satisies 79 y Ψ z =0 Thereore the two step method to obtain the inal estimate ˆμ gives the same results as a one shot approah o minimizing y z Appendix B: The Coarse-Sale EnKF Algorithm Run the simulation model up to a partiular observation time or entire ensemble to get predited samples: N i e N Ψ A = Ψ Ψ Ψ e = i ) Step : Using measured water ut data y with variane R get updated ensemble: N i e Ψ = Step Find ensemble mean (Equation (4)) Ψ Step Subtrat deviation rom mean N A '= b b b e i i b = Ψ Ψ Step 3 Apply H to eah olumn o A ' to get S = HA ' ie simply pik the water ut deviations in A ' Step 4 or i = N e iid Sample i ν 0R i i y = y ν = N R ν ν ν e N D = d d d e i = i i i d y W ; W is predited water ut or eah ensemble member end or / Step 5 Compute SVD SR = XLX R Get ˆ retaining irst ew singular values whih explain most variability in orresponding let singular vetors: X L Step 6 Update ensemble: Eqnuation (7) N e A Ψ Ψ Ψ T ˆ L ˆ ˆ T A A AS X XLD ) Step : Using oarse-sale data z with variane Q get updated ensemble: N i e Ψ i= Step Compute oarse-sale ensemble predition: i i u UΨ i N e Ne i Step Coarse-sale mean: μ '= i= N u e Step 3 Coarse-sale deviations: N S '= s s s e i i s = u μ ' Step 4 Repeat Step 4 using oarse-sale mea- i

16 80 surement or i = N e iid Sample i ω 0R i i z = z ω / N Q = ω ω ω e N D '= d d d e i i i d = z u end or / Step 5 Compute SVD S' Q = XLX R Get ˆ and X L as in step 5 Step 6 Compute ine-sale mean: Ne i μ'= i= N Ψ e Step 7 Compute ine-sale deviations: N A = b b b e i i b Ψ μ Step 8 Update ensemble: N e A Ψ Ψ Ψ T ˆ ˆ ˆ T A A A S X X D Remark : L Note that steps 6 and 7 in above algorithm approximate the intermediate ine-sale error ovariane L Remark : P A A N e Steps -3 aomplish 3 S'= UA Note that the above algorithm is independent o the hoie o upsaling proedure and also we an use the same algorithm or dierent kinds o oarse-sale observed data (i available) Remark 3: Note that the above oarse-sale onstrained EnKF algorithm an be readily extended to inorporate data at multiple oarse sales with appropriate upsaling proedure in U To elaborate i we had another independent data at a sale dierent rom z we use the estimates ( N i e Ψ ) obtained using z as intermediate solution i= repeat Step to assimilate the data at another sale T 3 As noted in [] this approah o aounting or nonlinear observations operator U works well as long as U is weakly nonlinear and a monotoni untion o model variables Ψ