SPE Copyright 2007, Society of Petroleum Engineers

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1 SPE Adaptve Multscale Streamlne Smulaton and Inverson for Hgh-Resoluton Geomodels V. R. Stenerud, SPE, Norwegan U. of Scence and Technology; V. Kppe, SINTEF ICT; A. Datta-Gupta, SPE, Texas A&M U.; and K.-A. Le, SINTEF ICT. Copyrght 2007, Socety of Petroleum Engneers Ths paper was prepared for presentaton at the 2007 SPE Reservor Smulaton Symposum held n Houston, Texas, U.S.A., February Ths paper was selected for presentaton by an SPE Program Commttee followng revew of nformaton contaned n an abstract submtted by the author(s). Contents of the paper, as presented, have not been revewed by the Socety of Petroleum Engneers and are subject to correcton by the author(s). The materal, as presented, does not necessarly reflect any poston of the Socety of Petroleum Engneers, ts offcers, or members. Papers presented at SPE meetngs are subject to publcaton revew by Edtoral Commttees of the Socety of Petroleum Engneers. Electronc reproducton, dstrbuton, or storage of any part of ths paper for commercal purposes wthout the wrtten consent of the Socety of Petroleum Engneers s prohbted. Permsson to reproduce n prnt s restrcted to an abstract of not more than 300 words; llustratons may not be coped. The abstract must contan conspcuous acknowledgment of where and by whom the paper was presented. Wrte Lbraran, SPE, P.O. Box , Rchardson, Texas U.S.A., fax Abstract A partcularly effcent flow solver can be obtaned by combnng a recent mxed multscale fnte-element method for computng pressure and velocty felds wth a streamlne method for computng flud transport. Ths multscalestreamlne method has shown to be a promsng approach for fast flow smulatons on hgh-resoluton geologc models wth multmllon grd cells. The multscale method solves the pressure equaton on a coarse grd whle preservng mportant fne-scale detals. Fne-scale heterogenety s accounted for through a set of generalzed, heterogeneous bass functons that are computed numercally by solvng local flow problems. When ncluded n the coarse-grd equatons, the bass functons ensure that the global equatons are consstent wth the local propertes of the underlyng dfferental operators. The multscale method offers a substantal gan n computaton speed, wthout sgnfcant loss of accuracy, when the multscale bass functons are updated nfrequently throughout a dynamcs smulaton. In ths paper we propose to combne the multscalestreamlne method wth a recent generalzed travel-tme nverson method to derve a fast and robust method for hstory matchng hgh-resoluton geologc models. A key pont n the new method s the use of senstvtes that are calculated analytcally along streamlnes wth lttle computatonal overhead. The senstvtes are used n the travel-tme nverson formulaton to gve a robust quaslnear method that typcally converges n a few teratons and generally avods much of the subjectve judgments and tme-consumng traland-errors n manual hstory matchng. Moreover, the senstvtes are used to control a procedure for adaptve updatng of the bass functons only n areas wth relatvely large senstvty to the producton response. The senstvtybased adaptve approach allows us to selectvely update only a fracton of the total number of bass functons, whch gves a substantal savngs n computaton tme for the forward flow smulatons. We demonstrate the power and utlty of our approach usng a smple 2D model and a hghly detaled 3D geomodel. The 3D smulaton model conssts of more than one mllon cells wth 69 producng wells. Usng our proposed approach, hstory matchng over a perod of 7 years s accomplshed n less than twenty mnutes on an ordnary workstaton PC. Introducton It s well known that geomodels derved from statc data only such as geologcal, sesmc, well-log and core data often fal to reproduce the producton hstory. Reconclng geomodels to the dynamc response of the reservor s crtcal for buldng relable reservor models. In the past few years, there have been sgnfcant developments n the area of dynamc data ntegraton through the use of nverse modelng Streamlne methods have shown great promse n ths regard Streamlne-based methods have the advantages that they are hghly effcent forward smulators and allow senstvtes of producton responses wth respect to reservor parameters to be computed analytcally usng a sngle flow smulaton Senstvtes descrbe the change n producton responses due to small perturbatons n reservor propertes such as porosty and permeablty and are a vtal part of many dynamc data-ntegraton processes. Our recent works on streamlne-based ntegraton of producton data were based on so-called generalzed travel tme nverson. 7,8 There are several advantages assocated wth travel-tme nverson of producton data. Frst, t s robust and computatonally effcent. Unlke conventonal ampltude matchng, whch can be hghly non-lnear, t has been shown that the travel-tme nverson has quaslnear propertes. 7,13 As a result; the mnmzaton proceeds rapdly even f the ntal model s not close to the soluton. Second, the travel tme senstvtes are typcally more unform between wells compared to ampltude senstvtes that tend to be localzed near the wells. Ths prevents over-correcton n the near-well regons. 13 Fnally, n practcal feld applcatons, the producton data are often characterzed by multple peaks (for example, tracer responses). Under such condtons, the travel-tme nverson can prevent the soluton from convergng to secondary peaks n the producton response. 7

2 2 SPE Even though streamlne smulaton provdes fast forward smulaton compared wth a full fnte-dfference smulaton n 3D, the forward smulaton s stll the most tme-consumng part of the hstory-matchng process. A streamlne smulaton conssts of two steps that are repeated: () soluton of a 3D pressure equaton to compute flow veloctes; and () soluton of 1D transport equatons for evolvng flud compostons along representatve sets of streamlnes, followed by a mappng back to the underlyng pressure grd. The frst step s referred to as the pressure step or the pressure solve and s often the most tme-consumng. Consequently, hstory matchng and flow smulaton are usually performed on upscaled reservor-smulaton models. 14 However, upscalng/downscalng may result n loss of mportant fnescale nformaton. Recently, several multscale methods for solvng the pressure equaton have proven to be a promsng alternatve to standard upscalng, both wth respect to accuracy and effcency These multscale methods are specally desgned to perform well when the underlyng parameters exhbt a multscale structure; that s, when the parameter values span several orders of magntude or the correlaton lengths of the heterogenety structures vary over several orders. Lke standard upscalng methods, the multscale methods compute coarse-scale pressure veloctes by solvng a reduced set of global flow equatons on a coarsened grd. These coarse-scale equatons are defned n terms of the solutons of a set of local, decoupled flow problems. However, unlke upscalng methods, whch only preserve the local flow n an averaged sense, multscale methods try to preserve the subscale varatons of the flow by fndng approxmate solutons that contan the most relevant fne-scale nformaton. As a result, multscale methods gve pressure and/or flow veloctes on both the coarse grd and on the orgnal fne grd. Multscale methods are prmarly targeted at dynamc flow smulatons, where the pressure needs to be computed repeatedly. Snce the temporal changes n the varable coeffcents n the pressure equaton are typcally moderate compared to the spatal varablty, t s seldom necessary to recompute the local flow problems each tme the pressure s updated. Each local flow problem s computed ntally as part of a preprocessng step (that s embarrassngly smple to parallelze) and typcally only updated f the local doman s swept by a strong front n the flud compostons or the global flow pattern changes sgnfcantly due to shut-n of wells, nfll drllng, well converson, etc. Hence, a pressure update typcally conssts of recomputng a few local flow problems and then solvng a global flow problem on the coarse grd. Ths means that one can obtan an approxmate soluton on the orgnal grd at the cost of solvng the same problem on a much coarser grd. In ths paper, we combne multscale-streamlne smulaton and streamlne-based hstory matchng n one effcent approach. For the pressure equaton n the forward model we wll apply a method we refer to as the mxed multscale fnte-element method (MsMFEM). 19,20 The method uses a two-grd approach consstng of a fne grd, on whch the geomodel s gven, and a coarse grd, on whch the global flow problem s solved. The blocks n the coarse grd are gven as connected collectons of cells from the fne grd. For each par of adjacent blocks n the coarse grd, a local flow problem s solved on the underlyng fne grd to obtan bass functons that are ncorporated nto a global system of equatons assocated wth the coarse grd. MsMFEM produces massconservatve solutons both on the coarse grd and on the underlyng fne grd, s flexble wth respect to grd representaton (geometry/topology), and has a rgorous mathematcal framework. For the hstory matchng we use the generalzed traveltme nverson method, whch has prevously been successfully appled to many feld cases. Central n ths method s the computaton of analytc streamlne senstvtes n terms of smple 1-D ntegrals along streamlnes. The senstvtes can be computed usng a sngle streamlne smulaton. The second novel dea n ths paper s a strategy based on senstvty thresholdng for reducng the workload for the forward smulaton and for the nverson process. Altogether, the analytc senstvtes are used for three purposes: () n the nverson method, () to reduce the computatonal complexty of the forward smulatons by reducng the number of local flow solves, and () to reduce computatonal complexty of the nverson process. The outlne of our paper s as follows. Frst, we dscuss the basc steps n our proposed approach and llustrate the hstorymatchng procedure usng a smple synthetc example. Second, we descrbe the multscale-streamlne flow smulaton and the hstory-matchng procedure. Thrd, we dscuss and demonstrate the mpact of selectve senstvty-based workload reducton. Fnally, we present a hgh-resoluton hstorymatchng example to demonstrate the effcency and the practcal applcablty of our method. Background and Illustraton of the Procedure Streamlne-based hstory matchng or nverse modelng utlzes streamlne-derved senstvtes to calbrate geomodels to dynamc data. The major steps nvolved n the proposed process are: () Multscale-streamlne flow smulaton to compute producton responses at the wells. () Quantfcaton of the msmatch between observed and computed producton responses va a generalzed travel tme. An optmal traveltme shft s computed by systematcally shftng the computed producton responses towards the observed data untl the cross-correlaton between the two s maxmzed. 8 () Computaton of streamlne-based analytc senstvtes of the producton responses (water-cuts) to reservor parameters, specfcally permeablty. (v) Updatng of reservor propertes to match the producton hstory va nverse modelng. We propose a senstvty-based thresholdng strategy to reduce the computatonal work for ths step. Ths four-step process s repeated untl a satsfactory match n producton data s obtaned. To reduce the computatonal workload for the forward smulaton, we propose to reuse bass functons n regons wth low senstvty to the producton responses. In the next sectons we wll dscuss the detals of the mathematcal formulaton behnd the multscale mxed fnteelement formulaton and the nverson method, and propose a senstvty-based strategy for selectve work reducton. However, for clarty of exposton, we frst llustrate the hstory matchng procedure usng a 2-D synthetc example.

3 SPE A Synthetc Example. Ths synthetc case (Case 1) nvolves reconstructon of a reference permeablty dstrbuton on a unform grd (Fg. 1), based on the observed water-cut producton hstory from a 9-spot pattern (Fg. 2). For the forward smulaton we apply the MsMFEM-streamlne smulator (to be descrbed below), wth the grd as the underlyng fne grd. We construct a unform coarse grd of dmenson 7 7 so that each coarse-grd block conssts of 3 3 subcells. Fgure 3 llustrates the two-grd approach for a slghtly more general case wth nonmatchng blocks n the coarse grd. The multscale smulator bascally works as follows: For each par of adjacent blocks n the coarse grd, a local flow problem s solved to obtan a local (multscale) bass functon assocated wth the correspondng nternal face n the coarse grd (see Fg. 4). The local bass functons are then ncorporated nto a global system of equatons defned on the coarse grd, whch s solved to obtan coarse-grd fluxes. Fne-scale flow veloctes are then obtaned by multplyng the coarse-grd fluxes wth the correspondng multscale bass functon and summng over all edges n the coarse grd. The flow was descrbed usng quadratc relatve permeablty curves wth an end-pont moblty rato of M=0.5. The water-cut responses obtaned from a flow smulaton of the reference permeablty feld are shown n Fg. 2 (wth 5% whte nose added). We treat these as the observed data. Next, startng from a homogenous ntal permeablty feld, we match the water-cut data va the generalzed travel-tme nverson. The permeablty for each fne-grd cell s treated as an adjustable parameter for ths example (a total of 441 parameters). A comparson of the ntal and fnal match of the water-cuts s shown n Fg. 2. Overall, the match to the producton data s qute satsfactory. Fgure 5 shows the reducton n tme-shft and ampltude resdual. The fnal permeablty dstrbuton s shown n Fg. 1. Clearly, the fnal permeablty model captures the large-scale trends of the reference permeablty feld. Mathematcal Formulaton Multscale Flow Smulaton. An mportant aspect of the proposed hstory-matchng algorthm s the use of a multscale mxed fnte-element method (MsMFEM) for the pressure equaton. Ths method belongs to a famly of multscale fnteelement methods, frst ntroduced by Hou and Wu. The basc 21 dea of the methods s to construct specal fnte-element bass functons that are adaptve to the local propertes of the ellptc dfferental operator. To ensure local mass conservaton on the coarse and fne grd, Chen and Hou ntroduced a multscale 19 method based on a mxed fnte-element dscretzaton. The method was later modfed by Aarnes to ensure local mass 20 conservaton also for blocks contanng source terms. In the current paper, we use a slghtly dfferent formulaton due to 22 Aarnes and Le. Governng Equatons. We consder ncompressble twophase flow of ol and water n a non-deformable permeable medum. For smplcty, we neglect the effects of gravty, compressblty and capllary forces. Further, we also assume for smplcty no-flow boundary condtons for the reservor. The flow equatons can be formulated as an ellptc equaton for the pressure p and the total velocty v, v = λ t k p, v = q (1) Here q s a source term representng njecton and producton wells, k s the absolute permeablty, and λ t = λ t (S w ) s the total moblty. We wll solve Eq. 1 for the fne-scale velocty feld v usng MsMFEM, for whch the detals wll be descrbed n the next subsectons. The velocty feld s used to obtan a streamlne dstrbuton. Along each streamlne the 3D transport equaton reduces to a 1D transport equaton wth the tme-of-flght as the spatal coordnate S w f w( Sw) + = (2) t τ Ths equaton s solved forward n tme along each streamlne usng front trackng. 23,24 Here the man advantage of ths method s that t s uncondtonally stable and therefore avods the usual CFL-constrant that would have put a severe lmtaton on the tme-step sze (.e., enforce the tme step to be smaller than what s preferred wth respect to accuracy). Mxed Fnte Elements. The mxed fnte-element formulaton of the flow equaton (Eq. 1) n a doman Ω seeks a par (v,p) n U V, such that Ω 1 v ( λ k) u dx p u dx = 0, u U,... (3) Ω Ω l v dx = ql dx, l V. Ω. (4) Here U and V are (fnte-dmensonal) functon spaces for pressure and velocty, respectvely. Now, lettng {Ψ } and {Φ k } be bases for U and V, respectvely, we obtan approxmatons v=σv Ψ and p=σp k Φ k, where the coeffcents v= {v } and p= {p k } solve a lnear system of the form B T C C v 0 = 0, p q..... (5) where B={b j }, C={c j } and q={q k } are defned by b c 1 j = Ψ ( λ k) Ψ j dx, (6) Ω k Ω = Φ Ψ dx, q Φ kq dx. k (7) k =.... (8) Ω Multscale Bass Functons. In a standard dscretzaton, the spaces U and V typcally consst of low-order pecewse polynomals. In multscale methods, U and V are gven by the soluton of local flow problems. For ncompressble flows, the actual pressure soluton s mmateral for the flow smulaton, and so only the velocty feld s needed. We wll therefore only ms construct an accurate multscale approxmaton space U for the velocty and use a standard approxmaton space V for pressure consstng of pecewse constant functons. Let {K m } be a parttonng of Ω nto mutually dsjont (fne) grd cells. Furthermore, let {T } be a coarse parttonng of Ω, n such a way that whenever K m T 0, then K m T (see

4 4 SPE Fg. 3). Let Γ j denote the non-degenerate nterfaces Γ j = T T j. For each Γ j, we assgn a bass functon Ψ j n U ms, and for each T we assgn a bass functon Φ n V. The bass functon Ψ j s obtaned by forcng a unt flow from block T to T j ; that s, by solvng a local flow problem n Ω j =T T j Ψ j = k Φ, t j Ψ j w ( x), = w j ( x), x T, λ... (9) x T, wth Ψ n=0 on the boundary of Ω j. To gve a unt flow from T to T j, the source terms w (x) are normalzed w ( ) ( ) ( ) x = W x. W ξ dξ. (10) T To ensure a conservatve approxmaton of v on the fne grd, we choose W =q for coarse blocks contanng a well. 20 For coarse blocks where q=0, we scale W accordng to the trace of the permeablty tensor 25 ;.e., we use trace ( ) ( k( x) ), f q( x) T = 0, W x =.... (11) q( x), otherwse. The local flow problems n Eq. 9 can be solved numercally by any consstent and conservatve method; here we use the standard two-pont flux-approxmaton scheme. The correspondng bass functons can be seen as generalzatons of the lowest-order Ravart-Thomas bass functons n a standard mxed method. 26 Fgure 4 llustrates the x-velocty bass functons n two dfferent cases. Ths completes the defnton of MsMFEM. Implementaton of the MsMFEM. We wll brefly descrbe some mplementaton aspects related to the effcency and generalty of MsMFEM. The mxed formulaton leads to an ndefnte global system (Eq. 5), whch may be more dffcult to solve effcently than the symmetrc postvedefnte (SPD) systems that typcally arse from standard dscretzaton methods. However, by reformulatng Eq. 3 and Eq. 4 to an equvalent, so-called hybrd, formulaton, t s possble to obtan an SPD system also for MsMFEM. Lke the ndefnte system n Eq. 5, the hybrd system wll be sparse because the bass functons have local support, and the soluton can be obtaned usng one of the effcent lnear solvers specalzed for sparse SPD systems. The hybrd 27 formulaton s descrbed n more detal by Aarnes et al. We note that n our current mplementaton, we solve the global system n Eq. 5 usng a drect sparse solver, snce we only deal wth moderately szed coarse systems. Most of the computatonal work n MsMFEM s assocated wth solvng the local flow problems defned by Eqs. 9 to 11, and the choce of soluton strategy for these equatons s crucal to the overall performance of the method. The local problems are usually small to moderately szed, and the resultng systems can be solved usng sparse drect or sparse teratve lnear solvers. The optmal choce of lnear solver typcally depends on the problem sze, and we recommend havng avalable a range of solvers tuned to dfferent system szes. Alternatvely, f one has access to a hghly effcent solver for large sparse systems, t may be benefcal to lump together several local problems to form a larger system. 1 j Solvng larger systems may be advantageous because the most effcent lnear solvers typcally requre an ntal setup phase. Regardless choce of soluton strategy, effcent parallelzaton s easy, snce the local flow problems are completely decoupled. In the examples presented n ths paper, we only use Cartesan grds. However, MsMFEM s flexble wth respect to the choce of both fne and coarse grds. Gven a fne-grd solver, bass functons can be defned for almost any collecton of connected fne-grd cells. 25 Recently, the method has been mplemented for (matchng) corner-pont and tetrahedral grds n 3D 27, and based on ths experence we are confdent that the methodology presented n the current paper s easly extended to corner-pont grd models. Producton-Data Integraton. In our approach, ntegraton of producton data s carred out usng a generalzed travel-tme nverson as descrbed by He et al. 8 Frst, the producton-data msmatch s determned by computng a generalzed traveltme msft for the water-cut at each producng well. Ths s accomplshed by shftng the computed water-cuts towards the observed data untl the correlaton between the two s maxmzed. The nverson algorthm smultaneously mnmzes the travel-tme msft for all the wells usng an teratve least-square mnmzaton algorthm (LSQR). 7,8 The basc underlyng prncples behnd the hstory-matchng algorthm are brefly as follows: Match the feld-producton hstory wthn a specfed tolerance. Ths s accomplshed by mnmzng the travel-tme msft for water-cut. Preserve geologcal realsm by keepng changes to the pror geologcal model mnmal, f possble. The pror model already ncorporates statc data (well and sesmc data) and avalable geologcal nformaton. Only allow for smooth and large-scale changes; the producton data has low resoluton and cannot be used to nfer small-scale varatons n reservor propertes. Formulaton of Inverse Problem. Mathematcally, ths algorthm leads to the mnmzaton of a penalzed msft functon consstng of the followng three terms 7,8 : Δ ~ t S m + β δm + β Lδm δ (12) Here Δ ~ t s the vector of generalzed travel-tme shfts at the wells, S s the senstvty matrx contanng the senstvtes of the generalzed travel tme wth respect to the reservor parameters, δm corresponds to the changes n the reservor propertes, and L s a second-order spatal dfference operator. The frst term ensures that the dfference between the observed and calculated producton response s mnmzed. The two remanng terms are standard regularzaton terms. The second term s a norm constrant that penalzes devatons from the ntal (pror) geologcal model and as such helps to preserve the geologcal realsm n the hstory match. The thrd term, whch s a roughness constrant that measures the regularty of the changes, s ntroduced to stablze the nverson. Physcally, t only allows for large-scale changes that are consstent wth the low resoluton of the producton data. The

5 SPE weghts β 1 and β 2 determne the relatve strengths of the pror model and the roughness term. The mnmum n Eq. 12 can be obtaned by an teratve least-squares soluton to the augmented lnear system ~ S Δt β 1I δm = 0... (13) β2l 0 Ths system s solved wth the teratve least-square mnmzaton algorthm, LSQR, 28 for whch the computatonal cost scales lnearly wth respect to the number of degrees-offreedom. 29 Fne-grd senstvtes close to zero are elmnated, whch makes the system more sparse and reduces the number of arthmetc operatons for the LSQR-teratons. In the next secton we wll dscuss an approach to further reduce the number of non-zero senstvtes based on thresholdng of coarse-grd senstvtes. In our mplementatons we focus on nvertng water-cut data. However, the generalzed travel-tme nverson method has earler been extended to compressble three-phase flow, so that water-cut and gas-ol-ratos are ncorporated jontly. 10 Water-Cut Senstvtes. A unque feature of streamlne methods s that the parameter senstvtes can be computed usng a sngle flow smulaton, leadng to very fast hstorymatchng or nverse-modelng algorthms. Moreover, because the senstvtes are smple ntegrals along streamlnes, the computaton tme scales very favorable wth respect to the number of grd cells, thus makng streamlnes the preferred approach for hstory matchng hghly-detaled geologcal models. For the sake of completeness, we fnally brefly descrbe the streamlne-based senstvty calculatons. The velocty of propagaton for a gven saturaton contour S w along a streamlne wll be gven by, τ t Sw df = ds w w...(14) and the arrval tme of the saturaton front wll be, df w a = τ.. (15) dsw t We use the above relatonshp to compute the senstvty of the arrval tme of the saturaton front based on the senstvty of the tme-of-flght. 7,8 Specfcally, the senstvty of the arrval tme of the saturaton front wth respect to reservor parameter m s computed as, τ ta = m m dfw ds w....(16) Here the senstvty of the tme-of-flght s computed analytcally from a sngle streamlne smulaton under the assumpton that the streamlnes do not shft because of small perturbatons n reservor propertes. For example, the tme-offlght senstvty wth respect to permeablty n grd cell, under the assumpton of the same permeablty for the whole grd cell, wll be gven by 7 τ τ s( ξ ) s( ξ ) τ = = dξ = dξ =,..(17) k k k k k Σ Σ where the ntegral s along the streamlne trajectory Σ and s(x) s the slowness defned as the recprocal of the total ntersttal velocty φ( x) μ φ( x) s( x) = =. (18) v( x) k( x) P Smlarly, the tme-of-flght senstvtes can be calculated wth respect to moblty or to the product of moblty and permeablty. Fnally, the senstvty of the shft n the generalzed travel tme Δ ~ t wth respect to reservor parameters s gven by Δ~ t 1 N t d = = a, m N 1 d a m. (19) where N d represents the number of observed data for a well. Fnally, we remark that the streamlne-based senstvty computaton has been extended to nclude gravty, changng feld condtons, and fractured reservors. 8,9 Senstvty-Based Selectve Work Reducton In ths secton, we dscuss how the senstvtes ntroduced above can be used to reduce the computatonal complexty of the hstory matchng wth neglgble loss n qualty of the derved match. To ths end, we use senstvtes to determne when to update and when to not update bass functons. Smlarly, we wll reduce the nverse system by only ncludng the senstvtes of fne-grd cells wthn coarse blocks havng a suffcently hgh average senstvty. These two strateges wll be descrbed n more detal below. Selectve Updatng of Bass Functons. To reduce the computatonal work for MsMFEM, we propose to only update bass functons n areas wth great producton-response senstvtes. The nverson method provdes senstvtes assocated wth the fne grd. To assocate a senstvty coeffcent wth each bass functon, we compute the arthmetc average of fne-grd senstvtes over the doman where the bass functon has support. To determne whch bass functons to update, one can ether: () use a predefned threshold for the senstvty values, or () update a predefned fracton of the bass functons. The frst approach s fully adaptve n the sense that the number of updated bass functons may change from teraton to teraton. However, ths approach requres (general) gudelnes for settng the threshold, whch may be easer to obtan by makng the senstvtes dmensonless. The second approach requres sortng of the senstvtes. The number of bass functons s equal to the number of edges/faces, whch scales wth the number of coarse-grd blocks, and sortng them s therefore a mnor concern, snce the number of operatons for sortng N numbers scales lke N logn. For our mplementatons we wll stck to the second approach. By nspectng Eq. 9, we notce that there are three factors that may requre the bass functons to be updated before a new pressure solve. Frst of all, we notce that f the absolute

6 6 SPE permeablty k changes, the bass functons wll change, too. In the current applcaton, the absolute permeablty wll typcally change (n certan regons) from one forward smulaton to the next. Secondly, f the well rate q changes, the source term w (x) wll change and hence bass functons wth support n well-blocks wll change. Fnally, f the total moblty λ t changes, due to changes n saturaton (or vscostes), the bass functons wll change. In the frst flow smulaton of the hstory-matchng procedure, we update all bass functons n every pressure step, because no senstvtes are yet avalable. (In a more sophstcated mplementaton, one would typcally have used another knd of ndcator to ensure that bass functons are only updated near the saturaton front 20,16 ). After the frst smulaton, the permeablty feld s updated by the nverson method. Snce the permeablty feld has changed, we should, at least n prncple, recalculate all bass functons for the frst pressure step of the next flow smulaton. For the subsequent pressure steps of the smulaton, we apply the proposed selectve updatng strategy. For the subsequent smulatons we repeat the strategy of the second smulaton. The approach descrbed n ths paragraph, when x% of the bass functons are updated dynamcally each tme step, s referred to as x% BF-DU (bass functons dynamcal update). Fnally, we remark that for 0% BF-DU we wll not update bass functons durng the frst flow smulaton because no senstvtes are then requred. Ths specal case devates from what we specfed above. An extended approach would be to reuse bass functons from the prevous forward smulaton n the coarse blocks where the absolute permeablty has undergone small or no changes n the last nverson step. We suggest to use averaged producton-response senstvtes to pck coarse blocks wth small changes (or generally coarse blocks that have lttle effect on the overall producton characterstcs). Alternatvely, one could use some knd of norm crtera to determne changes to the absolute permeablty. We wll refer to ths strategy, where x% of the bass functons are updated ntally and the remanng (100-x)% are kept from the prevous flow smulaton, as x% BF-IU (bass functons ntal update). In the followng, we wll only use one of these two technques at a tme, although they may n prncple be combned. Selectve Reducton of the Inverson System. Snce the water-cut data contan lmted nformaton about fne-scale varatons, t can be advantageous to avod nvolvng areas of low senstvty n the nverson, and nstead focus on resolvng large-scale structures n areas wth hgher senstvtes. We therefore propose to elmnate fne-scale senstvtes from the LSQR-system (Eq. 13) n areas of low senstvty, whch wll also reduce the computatonal work n the nverson process. To determne the areas of low and hgh senstvty, we use a smlar procedure as for the selectve updatng of bass functons. That s, for each coarse block we compute a senstvty coeffcent by arthmetc averagng of the fne-grd senstvtes already provded by the nverson method. Then we ntroduce a threshold and only nclude the fne-scale senstvtes assocated wth cells nsde coarse blocks havng an average senstvty above the gven threshold. The coarse blocks that are elmnated n ths process wll usually manly contan cells wth zero or low senstvty. The constrants nvolved n Eq. 12 are mportant for the elmnaton of coarse blocks to work. As for the thresholdng of bass functons, we can ether use a predefned threshold for the senstvty values or a predefned fracton of coarse blocks; here we use the second approach. Henceforth, keepng y% of the coarse blocks s referred to as y% CB. It should be noted that elmnatng fne cells for a fracton of the coarse blocks havng low senstvty wll not necessarly decrease the number of fne-grd senstvtes n the nverse system by the same fracton. The reason s cells wth zero or small senstvty are already elmnated, and such fne-grd senstvtes are more lkely represented n coarse blocks wth low senstvty. Impact of Selectve Work Reducton To nvestgate the accuracy of the proposed selectve work reducton, we apply t to the synthetc 9-spot case presented earler n ths paper (Case 1). We wll stll refer to ths case as Case 1 even though we wll vary some parameters and strateges for selectve work reducton. Frst, we nvestgate the effect of the proposed strategy for selectve dynamcal updatng of bass functons (BF-DU) for the forward smulaton. To ths end, we compare results obtaned by updatng 100%, 75%, 50%, and 25% of the bass functons, selected accordng to the assocated coarse-grd senstvty coeffcents. (Here all bass functons are recomputed n the frst step of each new forward smulaton). Table 1 shows the correspondng reductons for msft n tmeshft and ampltude after sx teratons for an unfavorable moblty rato (M=10) and two favorable moblty ratos (M=0.2 and M=0.5). In addton, the table reports the average dscrepancy between the reference and derved permeablty feld measured by 1 N reference derved log( k ) log( k ).... (20) = 1 N Reductons n msft for each teraton n the hstory-matchng algorthm are shown n Fgs. 6 and 7 for M=10 and M=0.2, respectvely. Smlarly, some of the derved permeablty felds from the hstory matches are shown n Fgs. 8 and 9. In general, the data are well matched for all reducton strateges. However, snce the nverson problem s ll-posed, a unque soluton cannot be expected. Indeed the permeablty felds obtaned for the cases wth unfavorable and favorable moblty ratos dffer even when updatng all bass functons. Judgng from Fg. 8, the derved permeablty felds for the unfavorable moblty rato do not seem to change much when reducng the number of dynamcally updated bass functons. Followng Aarnes 20, one can argue that t s n general qute safe to reduce the number of dynamcally updated bass functons for unfavorable flow cases, snce these are characterzed by weak shocks and mostly smooth varatons n the total moblty. For the favorable moblty rato (M=0.2), the derved permeablty felds seem to change more by reducng the fracton of bass functons updated. In ths case, the flow wll generally have strong saturaton fronts, whch nduce major changes n the bass functons as the leadng water fronts move through the correspondng grd blocks.

7 SPE To further reduce the number of bass functon computatons we wll try to apply the extended approach, n whch we reuse bass functons from the prevous forward smulaton, for M=0.2, M=0.5 and M=10. We go drectly to the extreme of keepng the bass functons from the frst tme step of the frst flow smulatons throughout the hstory-matchng procedure. In other words, no updatng of bass functons at all; that s 0% BF-DU and 0% BF-IU. The correspondng results are reported n Table 1, and Fg. 10 shows the resultng permeablty felds for moblty ratos M=0.2 and M=10. The qualty of the hstory-matchng procedure does not seem to declne dramatcally by not updatng bass functons at all for ths case. Fnally, we nvestgate the effect of the proposed strategy for selectve reducton of the nverse system. To ths end, we consder the case wth moblty ratos M=0.2, M=0.5 and M=10, and keep the fne-grd senstvtes correspondng to 100%, 75%, and 50% of the coarse-grd blocks, selected by thresholdng the averaged senstvtes n the coarse grd. Keepng 100% for M=0.5 corresponds to the hstory matchng performed for the 9-spot case ntally n ths paper. All the fractons matched the data well (see Table 1 and Fg. 11). The derved permeablty felds for M=0.5 for the three strateges 100% CB, 75% CB, and 50% CB are shown n Fgs. 1, 12a, and 12b, respectvely. The derved permeablty felds do not change much when reducng the coarse-block fracton to 50%, but gong down to 25% gave a non-realstc permeablty feld. We also note that for some cases, the selectve reducton of the nverse system resulted n a slghtly slower convergence for the nverson. The method converged to the same resdual level as wthout selectve work reducton, but the nverson requred one or two addtonal teratons, thus resultng n ncreased total computaton tme. Even though the selectve reducton of the nverson system can result n a slghtly slower convergence, our experments demonstrate robustness for the generalzed travel-tme nverson. Fnally we nvestgate the reducton strategy for the forward smulaton (not extended approach) and the nverson smultaneously for the case wth moblty ratos M=0.2, M=0.5 and M=10. We consder the combnaton of 50% BF-DU wth 75% and 50% CB. The two cases matched the data well (see Table 1 and Fg. 11). Further, the derved permeablty felds for M=0.5 are shown n Fgs. 12c and 12d. The derved permeablty felds do not change much by ncludng selectve updatng of bass functons (see Table 1). Hstory Matchng a Full 3D Geomodel In ths secton we demonstrate the feasblty of the approach for feld studes by applcaton to a hgh-resoluton 3-D example (Case 2). As mentoned before, streamlnes and the tme-of-flght are used to compute the senstvty of the producton data wth respect to reservor parameters as descrbed above. In ths feld-scale example, water-cuts were matched to update the reservor permeablty dstrbuton usng the MsMFEM-streamlne smulator for the forward smulaton. Model Descrpton. The geomodel conssts of a fne grd wth cells, whch gves a total of 1,048,576 grd cells, each of sze m. The fne-grd cells are collected nto a unform coarse grd, so that each coarse block conssts of cells n the fne grd. All the cells are treated as actve. The permeablty s log-normally dstrbuted wth a mean of 2.2 md, a mnmum of md and a maxmum of 79.5 md (see Fg. 13b). The correlaton length n the x- and y- drectons s about 270 meters, and about 90 meters n the z- drecton. For our purposes, ths permeablty feld was used as a true or reference model to generate producton hstory from flow smulaton. To generate our reference producton data we used the standard two-pont flux-approxmaton (TPFA) fntevolume method drectly on the fne grd. A total number of 32 njectors and 69 producers were ncluded n the smulaton model (see Fg. 14). All the wells are vertcal and ntersect all layers. The producton hstory conssts of 2475 days of water-cut data from the 69 producers (Fg. 15). The water njectors were njectng at constant total reservor volume rate of 1609 bbl/day, and each producer was producng wth constant reservor volume rate fulfllng the total vodage rate. For each smulaton, we used 15 pressure steps of length 165 days. Further, quadratc relatve permeablty curves and end-pont moblty rato of M=5 were used. Producton Data Integraton. To generate an ntal permeablty model, we treat the permeablty values n the well-blocks of the reference model as known data. By condtonng on the well data, sequental Gaussan smulaton was used to generate multple realzatons of the permeablty model. 30 In the followng we wll manly consder three approaches: MsMFEM [full], MsMFEM [reduced] and TPFA. The two frst approaches are multscale approaches, whle the last one smulates drectly on the fne grd. Further, for the frst and the last approach no selectve work reducton occurs. For MsMFEM [reduced] the extended approach for reducng the number of bass functon computatons s appled. For each new forward smulaton, the bass functons are sorted accordng to average senstvtes and for the lowest 50%, bass functons are kept from the prevous flow smulaton. For the remanng 50%, the bass functons are updated once before the frst pressure solve. Moreover, selectve reducton of the nverse system s used keepng fne-grd senstvtes for 50% of the coarse blocks. In other words: MsMFEM [full] = 100% DU + 100% IU + 100% CB, MsMFEM [reduced] = 0% DU + 50% IU + 50% CB. Fgure 16 and Table 2 show the convergence of the nverson algorthm. In sx teratons, all msfts n tme-shft and ampltude for the water-cut dropped apprecably for the three approaches. Reference, ntal, and matched water-cut curves are shown n Fg. 15 for a few selected producers. Some of the wells had a qute good match ntally, and at the end of the hstory matchng all wells had a qute satsfactory match. Fgure 13 compares the ntal and the reference permeablty models wth the updated (derved) models. The scale s logarthmc and the mnmum permeablty s md and the maxmum s md. The three approaches gave almost dentcal derved permeablty felds. Therefore,

8 8 SPE just one of the derved permeablty felds (for MsMFEM [full]) s pcked for closer nspectons. From a casual look, t s hard to dscern the changes made to the ntal model. Ths s because the nverson algorthm s desgned to preserve the geologc contnuty and the ntal geologc features to the maxmum possble extent. However, a careful comparson reveals many dfferences between the ntal and the updated geologc models. Next, we examne f the changes made to the ntal model are consstent wth the reference permeablty model. Fgure 17 shows the dfferences between the updated and ntal permeablty model. These dfferences represent changes made. Ths s to be compared wth the changes needed, whch s the dfference between the reference and the ntal permeablty model. We see that there s clearly close agreement, partcularly n regons where the permeablty needs to be reduced (negatve changes). As mght be expected, there are also some dscrepances. Many of the wells had a good match ntally even though the permeablty felds dffer. Because the water-cut data curves are a result of the total flow pattern between a producer and one or more njectors, ths data source may have lmted spatal nformaton. Some of the changes occur n correct horzontal poston, but ncorrect vertcal poston. Ths can occur because the water-cut data has no vertcal spatal resoluton. Fnally, t s worth pontng out that ths nverson problem s hghly ll-posed, and therefore a varety of possble solutons exst. Table 2 shows average dscrepances between the reference and the derved permeablty felds (see Eq. 20) for TPFA, MsMFEM [full], MsMFEM [reduced]. The average dscrepances ndcate that the hstory-matchng procedure s stable wth respect to the selectve work-reducton strateges. We have also nvestgated some other selectve work-reducton strateges, and the results wth respect to both msft and average dscrepances turn out to be as stable as for Case 1. To sum up, the changes made to the permeablty feld preserved the geologc realsm, were mostly n accordance wth the changes needed (see Fg. 17), and resulted n satsfactory match of the water-cut data. Further, the dfferent strateges for selectve work-reducton turned out to gve stable results wth respect to changes made and msft (see Table 2). Computatonal Effcency. Fnally, we wll assess the effcency of our multscale method compared to a standard streamlne method usng a TPFA pressure solver. To ths end, we consder two dfferent computers runnng Lnux: PC 1 s a laptop PC wth a 1.7 GHz Intel Dothan Pentum M processor, 2Mb cache and 1.5 Gb memory, PC 2 s a workstaton wth a 2.4 GHz Intel Core 2 Duo, 4Mb cache, and 3 Gb memory. Table 2 reports smulaton tmes observed on the two computers. Here the total smulaton tme ncludes tme for nverson, IO, and seven forward smulatons, each wth ffteen pressure steps. Smlarly, we report the total tme for the pressure solves and the transport solves (ncludng mappngs and tracng of streamlnes). When all bass functons are updated n all steps, the multscale solver s, as expected, about 25% slower than TPFA wth an optmal algebrac multgrd (AMG) solver on the laptop (PC1). On the other hand, the memory requrements for MsMFEM are qute low and ths solver could easly have been run on larger models, as opposed to the TPFA methods, for whch the AMG solver almost exceeded the avalable memory. Moreover, on hghly skewed, non-cartesan grds (e.g., corner-pont grds), MsMFEM uses a much better spatal dscretzaton 27 and wll therefore gve more accurate predctons of flow. The comparson of TPFA and MsMFEM [full] s not very nterestng on the workstaton (PC2). Due to an mmature compler for the partcular hardware, we were not able to optmze the drect solver used to compute bass functons, whle the AMG solver could be (almost) fully optmzed by usng a vendor-specfc compler. The runtmes for the pressure solves (and the total runtme) on PC2 are therefore somewhat hgher than expected, and wll probably mprove sgnfcantly when a more mature compler becomes avalable n a few months. By MsMFEM [reduced], we were able to reduce tme for pressure solves by about 80% on both computers. In MsMFEM [reduced] the bass functons to be reused were read from fle. Slow dsc access on the laptop therefore prevented a further reducton n runtme. The workstaton, on the other hand, had a faster dsc, but further reductons n runtme were prevented by the unoptmzed lnear solver (as dscussed above). Reducton of the nverse system was expected to have a very small effect on the runtme, snce a fully optmzed complaton on a GHz processor gves a floatng-pont performance that would make the reduced number of arthmetc operatons nsgnfcant compared to other knds of operatons, whch ndeed s consstent wth what we observe n Table 2. However, the results from the reducton of the nverse system ndcate robustness for the generalzed traveltme nverson method. Fnally, to speed the method further up, and to make our smulatons comparable to state-of-the-art commercal streamlne solvers, we apply a method for mproved mass conservaton for streamlne smulaton proposed by Kppe et al. 31 Usng ths method, the total number streamlnes could be reduced from to , thereby reducng the tme for the transport solves by 80%. Altogether, ths meant that the full hstory match could be performed n an mpressve runtme of 17 mnutes on the workstaton (PC2) and 36 mnutes on the laptop (PC1)! For the workstaton there s an obvous potental for further mprovements by usng a better compler. Moreover, on Core 2 Duo processors one should also explot the natural parallelsm n updatng bass functons and n the streamlne computatons. Summary and Conclusons A novel approach to hstory matchng usng multscalestreamlne smulaton and analytc senstvtes s presented. The power and utlty of our proposed approach s demonstrated usng both a synthetc and a feld-scale example. The synthetc case ncludes matchng of water cut from a 9- spot pattern and s used to valdate the method. The feld-scale example conssts of more than a mllon grd cells. Startng

9 SPE wth a pror geomodel, producton data were ntegrated usng a generalzed travel tme nverson. The entre hstory matchng process took less than 40 mnutes usng a laptop PC and about 17 mnutes usng an ordnary workstaton PC. The permeablty changes were found to be reasonable and geologcally realstc. Some specfc conclusons from ths paper can be summarzed as follows. 1. A multscale-streamlne flow smulator was used for hstory matchng by generalzed travel-tme nverson. 2. By utlzng the producton-response senstvtes provded by the generalzed travel-tme nverson, we were able to reduce the total workload for the multscale smulator consderably and stll preserve the accuracy of the flow smulaton. 3. By utlzng the producton-response senstvtes, we were able to selectvely reduce the number of nonzero senstvtes n the nverse system consderably wthout reducng the accuracy of the producton data ntegraton. Ths demonstrates robustness for the generalzed travel-tme nverson. 4. The approach proved applcable and effcent for a hgh-resoluton reservor model. Acknowledgements The research of Stenerud was funded by the Uncertanty n Reservor Evaluaton (URE) program at the Norwegan Unversty of Scence and Technology. The research of Kppe and Le was funded by the Research Councl of Norway under grant number /S30. We wsh to thank Dr. Yalchn Efendev for helpful dscussons. Nomenclature v = velocty p = pressure l,u = test functons V,U = functon spaces K = fne grd cells/elements T = coarse grd blocks/elements Ω = doman Γ = coarse block nterface n = unt normal vector Ψ = bass functon velocty Φ = bass functon pressure q = total rate (source/snk) f w = fractonal flow functon (water) S w = saturaton of water k = absolute permeablty λ t = total moblty M = end-pont moblty rato m = reservor parameter N d = number of data ponts N = number of grd cells Subscrpts ms = multscale References 1. Anteron, F., Karcher, B., and Eymard, R. Use of Parameter Gradents for Reservor Hstory Matchng, paper SPE presented at the 1989 SPE Symposum on Reservor Smulaton, Houston 6-8 February. 2. Landa, J.L., Kamal, M.M., Jenkns, C.D., and Horne, R.N.: Reservor Characterzaton Constraned to Well Test Data: A Feld Example, paper SPE presented the 1996 SPE Annual Techncal Conference and Exhbton, Denver 6-9 October. 3. Wu, Z., Reynolds, A. C., and Olver, D.S.: Condtonng Geostatstcal Models to Two-Phase Producton Data, SPE Journal (June 1999) 4(2), Olver, D. S., Reynolds, A. C., B, Z., and Abacoglu, Y., Integraton of Producton Data nto Reservor Models, Petroleum Geoscence (2001) 7, S65-S Res, L.C., Hu, L.Y., and Eschard, R.: Producton Data Intergraton Usng a Gradual Deformaton Approach: Applcaton to an Ol Feld (Offshore Brazl), paper SPE presented at the 2000 Annual Techncal Conference and Exhbton, Dallas, Texas, 1-4 October. 6. Sahn, I., and Horne, R.: Multresoluton Wavelet Analyss for Improved Reservor Descrpton, SPE Reservor Evaluaton & Engneerng (February 2005), Vasco, D.W., Yoon, S., and Datta-Gupta, A.: Integratng Dynamc Data Into Hgh-Resoluton Models Usng Streamlne-Based Analytc Senstvty Coeffcents, SPE Journal (December 1999), He, Z., Datta-Gupta, A., and Yoon, S.: Streamlne-Based Producton Data Integraton wth Gravty and Changng Feld Condtons, SPE Journal (December 2002) 7(4), Al-Harb, M., Cheng, H., He, Zhong, and Datta-Gupta, A.: Streamlne-based Producton Data Integraton n Naturally Fractured Reservors, SPE Journal (December 2005), Cheng, H., Oyernde, D., Datta-Gupta, A., and Mllken, W.: Compressble Streamlnes and Three-Phase Hstory Matchng, paper SPE presented at the 2006 SPE/DOE Symposum on Improved Ol Recovery, Tulsa, Aprl. 11. Mllken, W. J., Emanuel, A. S., and Chakravarty, A.: Applcaton of 3-D Streamlne Smulaton to Assst Hstory Matchng, paper SPE presented at the 2000 Annual Techncal Conference and Exhbton, Dallas, Texas, 1-4 October. 12. Wang, Y., and Kovscek, A. R.: A Streamlne Approach to Hstory Matchng Producton Data, SPE Journal (2000) 5, Wu, Z., and Datta-Gupta, A.: Rapd Hstory Matchng Usng a Generalzed Travel Tme Inverson Method, SPE Journal (June 2002), Chrste, M.A., and Blunt, M.J.: Tenth SPE Comparatve Soluton Project: A Comparson of Upscalng Technques, SPE Reservor Evaluaton & Engneerng (August 2001), Gauter, Y., Blunt, M.J., and Chrste, M.A.: Nested Grddng and Streamlne-Based Smulaton for Fast Reservor Performance Predcton, Computatonal Geoscences (1999) 3, No. 3-4, Jenny, P., Lee, S.H., and Tchelep, H.A.: Adaptve Multscale Fnte-Volume Method for Multphase Flow and Transport n Porous Meda, Multscale Modelng and Smulaton (2004) 3, No. 1,

10 10 SPE Arbogast, T., and Bryant, S. L.: A Two-Scale Numercal Subgrd Technque for Waterflood Smulatons, SPE Journal (December 2002), Aarnes, J.E., Kppe, V., and Le, K.-A.: Mxed Multscale Fnte Elements and Streamlne Methods for Reservor Smulaton of Large Geomodels, Advances n Water Resources (2005) 28, No. 3, Chen, A., and Hou, T.H.: A Mxed Multscale Fnte Element Method for Ellptc Problems wth Oscllatng Coeffcents, Mathematcs of Computaton (2002) 72, No. 242, Aarnes, J.E.: On the Use of a Mxed Multscale Fnte Element Method for Greater Flexblty and Increased Speed or Improved Accuracy n Reservor Smulaton, Multscale Modelng and Smulaton (2004) 2, No. 3, Hou, T.Y. and Wu, X.-H.: A Multscale Fnte Element Method for Ellptc Problems n Composte Materals and Porous Meda, Journal of Computatonal Physcs (1997) 134, No. 1, Aarnes, J.E., and Le, K.-A.: Toward Reservor Smulaton on Geologcal Grd Models, Proc., 9 th European Conference on the Mathematcs of Ol Recovery, Cannes, France, 30 August - 2 September, (2004), B Bratvedt, F., Bratvedt, K., Buchholz, C.F., Holden, L., Holden. H., Rsebro, N.H.: A New Front-Trackng Method for Reservor Smulaton, SPE Reservor Engneerng (February 1992) 7, Holden, H., and Rsebro, N.H.: Front Trackng for Hyperbolc Conservaton Laws, Sprnger-Verlag New York Inc. (2002), ISBN Aarnes, J.E., Krogstad, S., and Le, K.-A.: A Herarchcal Multscale Method for Two-Phase Flow Based upon Mxed Fnte Elements and Nonunform Coarse Grds, Multscale Modelng and Smulaton (2006) 5, No. 2, Ravart, P.-A., and Thomas, J.M.: A Mxed Fnte Element Method for 2 nd Order Ellptc Problems, Mathematcal Aspects of Fnte Element Methods (Proc. Conf., Consglo Naz. Delle Rcerche (C.N.R), Rome, 1975), Lecture Notes n Mathematcs, Sprnger, Berln (1977) 606, Aarnes, J.E., Krogstad, S., and Le, K.-A.: Multscale Mxed/Mmetc Methods on Corner-Pont Grds, Computatonal Geoscences, submtted. 28. Page, C.C., and Saunders, M.A.: LSQR: An Algortm for Sparse Lnear Equatons and Sparse Least Squares, ACM Transactons on Mathematcal Software (March 1982) 8, No. 1, Vega, L., Rojas, D., and Datta-Gupta, A.: Scalablty of the Determnstc and Bayesan Approaches to Producton- Data Integraton nto Reservor Models, SPE Journal (September 2004) 9(3), Deutsch, C.V., and Journal, A.G.: GSLIB Geostatstcal Software Lbrary and User s Gude, Oxford Unversty, (1998). 31. Kppe, V., Hægland, H., and Le, K.-A.: A Method to Improve the Mass Balance n Streamlne Methods, paper SPE presented at the SPE Annual Reservor Smulaton Symposum 2007, Houston, February. Table 1 Case 1: Reducton n percent for msft n tme-shft (T) and ampltude (A), and reducton n average dscrepancy n log permeablty (Δlog(k)). The results are presented for dfferent strateges for selectve work reducton and dfferent moblty ratos M. The frst row shows the results for the ntal permeablty feld. Method M=0.2 M=0.5 M=10 DU IU CB T A Δlog(k) T A Δlog(k) T A Δlog(k) Intal % 100% 100% % 100% 100% % 100% 100% % 100% 100% % 100% 75% % 100% 50% % 100% 75% % 100% 50% %* 0% 100% *Does not update bass functons durng the frst flow smulaton Table 2 Case 2: Reducton n percent for msft n tme-shft (T) and ampltude (A). Reducton n average dscrepancy n log permeablty (Δlog(k)). The total smulaton tme for the hstory-matchng procedure for our mplementatons on two dfferent computers: a 1.7GHz laptop (PC 1), and a 2.4GHz workstaton (PC 2). The CPU-tme for the two computers spent on the pressure (velocty) solutons and the transport solutons. The frst row shows the results for the ntal geomodel. T A Δlog(k) Total smulaton Total CPU-tme: tme (Wall clock) Pressure Transport PC1 PC2 PC1 PC2 PC1 PC2 Intal TPFA h 12mn 1h 04mn 1h 02mn 33mn 54mn 28mn MsMFEM [full] h 42mn 2h 29mn 1h 17mn 1h 54mn 1h 06mn 32mn MsMFEM [reduced] h 34mn 43mn 9mn 7mn 1h 07mn 32mn MsMFEM [reduced - SL] mn 17mn 9mn 7mn 12mn 6mn

11 Fg. 1 Case 1: Reference and fnal permeablty felds (left and rght, respectvely) for moblty rato M=0.5. Fg The x-component of the velocty bass functon assocated wth an edge/face between two blocks of dfferent sze for a homogeneous and a heterogeneous permeablty feld, respectvely. Fg. 5 Case 1: Reducton of msft n tme-shft and ampltude of the water-cut. Moblty rato M=0.5. Fg. 6 Case 1: Reducton of msft n tme-shft and ampltude of the water-cut usng selectve updatng of bass functons. Moblty rato M=10. Fg. 2 - Case 1: water-cut match. Fg. 7 Case 1: Reducton of msft n tme-shft and ampltude of the water-cut usng selectve updatng of bass functons. Moblty rato M=0.2. Fg. 3 - A general coarse grd overlyng a unform fne grd wth the gray area gvng support of bass functon Ψ j, whch s assocated wth the edge/face ndcated by the red lne. Fg. 8 Case 1: Derved permeablty feld usng selectve updatng of bass functons. Updatng 100% (left) and 25% (rght) of the bass functons. Moblty rato M=10. The reference permeablty feld s shown n Fg. 3a.

12 12 SPE Fg. 11 Case 1: Reducton of msft n tme-shft and ampltude of the water-cut usng selectve work reducton also for the nverson system. Moblty rato M=0.5. Fg. 9 Case 1: Derved permeablty feld usng selectve updatng of bass. Updatng 100%, 75%, 50% and 25% of the bass functons (left to rght from top), respectvely. Moblty rato M=0.2. The reference permeablty feld s shown n Fg. 3a. a) 100% DU + 75% CB b) 100% DU + 50% CB Fg. 10 Case 1: Derved permeablty feld wthout updatng bass functons throughout the hstory matchng procedure for M=0.2 (left) and M=10 (rght). c) 50% DU + 75% CB d) 50% DU + 50% CB Fg. 12 Case 1: Derved permeablty feld usng selectve work reducton also for the nverson system. Moblty rato M=0.5. The reference permeablty feld s shown n Fg. 3a.

13 SPE (a) Intal permeablty (b) Reference permeablty (c) Derved permeablty (MsMFEM [full]) (d) Derved permeablty (MsMFEM [reduced]) (e) Derved permeablty (TPFA) Fg. 13 Case 2: Intal, derved and reference permeablty felds. Fg. 14 Case 2: Well confguraton for the geologc model example. The symbol x represents a producer whle the symbol o represents an njector.

14 14 SPE Fg. 15 Case 2: Water-cut match for 12 of the 69 producton wells ncluded n the hstory match of the geologc model (MsMFEM [full]). For each plot the sold red lne, the dash blue and the dashed purple lne represents the reference, the ntal and the updated water-cut curve, respectvely. (a) Shft-tme Msft (days). (b) Ampltude Msft. Fg. 16 Case 2: Reducton of msft n tme-shft and ampltude of the water-cut for hstory matchng of geologc model. Forward smulaton: MsMFEM [full] (blue sold curve), MsMFEM [reduced] (red dashed curve) and TPFA (black dash-dotted curve).

15 SPE a) Derved-Intal Permeablty Dfference (Layer 5-8) b) True-Intal Permeablty Dfference (Layer 5-8) c) Derved-Intal Permeablty Dfference (Layer 29-32) d) True-Intal Permeablty Dfference (Layer 29-32) Fg. 17 Case 2: Comparson of the derved-ntal permeablty dfference and the true-ntal permeablty (MsMFEM [full]).