A Workflow for Spatial Uncertainty Quantification using Distances and Kernels

Transcription

1 A Workflow for Spatal Uncertanty Quantfcaton usng Dstances and Kernels Célne Schedt and Jef Caers Stanford Center for Reservor Forecastng Stanford Unversty Abstract Assessng uncertanty n reservor performance requres the analyss of a large number of parameters. To capture the possbly wde range of uncertanty n flow response, a large set of realzatons needs to be processed. Geostatstcal algorthms can rapdly provde multple, equally probable realzatons. However, due to the large computatonal demand of flow smulatons, only a small number of realzatons can be smulated n practce. Statc propertes of the realzatons are therefore often used for rankng realzatons. Tradtonal rankng technques for selecton of P, P5 and P9 flow responses are hghly dependent on the statc property used (e.g. OOIP). In ths paper, we propose to parameterze the spatal uncertanty represented by a large set of geostatstcal models through a dstance functon measurng dssmlarty between any two geostatstcal realzatons. The dstance functon allows, through mult-dmensonal scalng, mappng the space of uncertanty to vsualze uncertanty. The dstance functon can be talored to the partcular problem, n ths case, flow responses. Ths mult-dmensonal space can be modeled usng kernel technques, such as kernel prncpal component analyss (KPCA). KPCA allows for the selecton of a subset of representatve realzatons contanng smlar propertes to the larger set. Wthout losng accuracy, producton decsons and strateges can then be performed usng flow smulaton on ths subset of realzatons wthout the need of extensve flow smulatons. A case study s presented, where spatal uncertanty of channel faces s modeled through multple realzatons generated usng a mult-pont geostatstcal algorthm and several tranng mages. Introducton Modelng spatal uncertanty for petroleum reservor s challengng due to the large number of parameters nvolved (possbly hundreds of parameters). Ths problem s compounded by the CPU demand of flow smulatons, makng tradtonal Monte Carlo approaches not always practcal. A wdely used approach for uncertanty quantfcaton s the expermental desgn technque. Expermental desgn ams at optmally selectng values of uncertan parameters n ther range of varaton, and then performng flow smulatons on the resultng models. From these flow smulatons, a proxy model of the flud flow smulator s bult, whch s a functon of the uncertan parameters. The proxy model allows the engneer to perform tradtonal Monte Carlo analyss of uncertanty.

2 However, one major drawback of expermental desgn technques s that they are often based on a smple lnear regresson and are thus not well suted for spatal varables or hgh-dmensonal problems. In addton, expermental desgn technques are not approprate for applcatons where there are many dscontnuous parameters (such as dfferng geologcal scenaros). An alternatve approach to quantfyng uncertanty s to examne a large set of model realzatons, and not ndvdual parameters. For problems of large dmensonalty, dealng wth the uncertanty on each reservor parameter separately s often not useful, snce many parameters are correlated, frequently n complex fashons. Moreover, ultmately, we are not nterested n uncertanty of ndvdual parameters, but n model realzatons bult from these parameters and responses predcted from those models. Rankng technques are tradtonally used to select realzatons that represent the P, P5 and P9 quantles of the flow responses of nterest. However, such technques are hghly dependent on the rankng property employed. Rankng s often based on a statc parameter, such as orgnal ol-n-place (OOIP) statc parameters may not be good ndcators of dynamc flow performance, often showng poor correlaton. The key dea of ths work s to propose a realzaton -based model of uncertanty parameterzed by dstances. Ths method employs a sngle parameter (the dstance) between any two model realzatons, whch can be talored to a partcular applcaton at hand. The am s to select a set of representatve reservor models by analyzng the propertes of the models as characterzed by the dstance, and fnally quantfy uncertanty wthn that set. The followng secton descrbes the methodology used n ths approach. An applcaton of the methodology on a synthetc case of channel faces s presented n Secton 3. We end ths paper by gvng some conclusons and dscusson of future work. Descrpton of Methodology The prncple of the methodology, llustrated for faces models, s descrbed n Fgure. Startng wth multple (N R ) realzatons generated usng any algorthm, a dssmlarty dstance matrx s constructed (Fgure a and b). Ths N R x N R matrx contans the dstance between any two model realzatons. The matrx s then used to map all realzatons nto a Eucldean space R (Fgure c), usng multdmensonal scalng. Each pont n ths map represents a realzaton. Snce n most cases the structure of the ponts n mappng space R s not lnear, we propose the use of kernel methods, n order to transform the Eucldean space R nto a new space F, called the feature space (Fgure d). The goal of the kernel transform s that ponts n ths new space behave more lnearly, so that standard lnear tools for pattern detecton can be used more successfully (Prncpal Component Analyss, cluster analyss, dmensonalty reducton, etc.). These tools allow the selecton of a few typcal ponts representng reservor models, among a potentally

3 very large set. Ths subset of models s small enough to allow uncertanty quantfcaton (e.g. P, P5, P9 quantles) through flow smulaton. Model Model δ δ 3 δ 4 δ 3 δ 4 3 δ 34 Model 3 Model δ δ δ 3 δ 4 δ δ δ 3 δ 4 δ 3 δ 3 δ 33 δ 34 δ 4 δ 4 δ 43 δ 44 (a) (b) (c) P,P5,P9 model selecton Φ (e) Φ (d) Fgure : Proposed workflow for uncertanty quantfcaton - (a) dstance between two models, (b) dstance matrx D, (c) models mapped n Eucldean space, (d) feature space, (e) pre-mage constructon Before gong nto more detal for each step of the methodology, we ntroduce some notaton: y, =,.., N R : model realzatons generated usng a mult-pont geostatstcal algorthm and several tranng mages N R : number of realzatons y N c : number of grd block of the realzatons : R δ j : dssmlarty dstance between realzatons y and y j x, =,.., N R : ponts n mappng space R representng realzatons y NC 3

4 . Measurement of Dssmlarty Dstance.. Defnton of dstance The frst step of the methodology s the defnton of a dssmlarty dstance between any two reservor models (Fgure a). The concept of smlarty between reservor models was ntroduced by Arpat (5), and Suzuk and Caers (6). The dstance s a way to determne how smlar two reservor models are n terms of geologcal propertes and producton response. The dstance between two models can be determned by classcal dstances that measure dfference n geometry such as the Hausdorff dstance, connectvty-based dstance, or by fast-flow smulaton (e.g. streamlne smulaton). Note that the concept of relatve dstance between two objects s dfferent from the concept of dfference n absolute statstcal summares. Let consder a smple example to llustrate ths last concept. Suppose that we are nterested n the dfferences n clmate of three ctes n the US: New York, San Francsco and Palo Alto, knowng only the dstances between the ctes and not ther geographcal locaton. To know how smlar the clmate s between each cty, we do not need to measure the temperature, humdty, and ranfall of each cty to know that the clmate s more smlar between San Francsco and Palo Alto ctes whch are very close (3 mles) - than between New York and San Francsco ctes are very far apart (934 mles). Applyng the analogy to reservor applcatons, we do not need to measure absolute reservor propertes/responses (such as OOIP and cumulatve ol producton) for each model to measure dssmlarty. We need only to defne a dstance between any two realzatons. The only requrement for ths dstance s a reasonable correlaton between the dssmlarty dstance between two reservor models and ther dfference n absolute flow responses. In the applcaton presented n Secton 3, we use results from streamlne smulaton (3DSL) to calculate the dstance. Note that the results from the streamlne smulaton are not used for uncertanty quantfcaton, but only to calculate the dstance. In the cases presented here, the dstance shows good correlaton wth dfferences n flow propertes generated by Eclpse... Constructon of a dssmlarty dstance matrx Gven a set of N R reservor models y and a dstance functon δ between any two models, a N R x N R dssmlarty dstance matrx D s constructed contanng the dstance measured between any two realzatons δ j (Fgure b). A vald dssmlarty matrx must satsfy both of the followng constrants: self-smlarty (δ = ) and symmetry (δ j = δ j ). Once the dstance matrx D s constructed, all the N R models are mapped nto a Eucldean space R usng multdmensonal scalng (MDS). 4

5 . Mult-Dmensonal Scalng (MDS) Mult-Dmensonal scalng (MDS) s a technque used to translate the dssmlarty matrx nto a confguraton of ponts n nd Eucldean space (Borg and Groenen, 997). The ponts n ths spatal representaton are arranged n such a way that ther Eucldean dstances correspond as much as possble (n least square sense) to the dssmlartes of the objects. Thus, one measurement of a successful MDS procedure s a good correlaton between the Eucldean dstance and the dssmlarty dstance. The algorthms desgned for analyzng a sngle dssmlarty matrx can be broadly dvded nto two basc types, classcal and nonmetrc MDS. Classcal MDS assumes that the dssmlarty matrx dsplays metrc propertes, such as dstances measured from a map. Thus, the dstances n classcal MDS space preserve the ntervals and ratos between the ponts as best as possble. Nonmetrc MDS s less restrctve, and only assumes that the order of the dssmlartes s meanngful. The order of the dstances n a nonmetrc MDS confguraton reflects the order of the dssmlarty as best as possble. Informaton such as ntervals and ratos between the ponts s not accounted for. Both classcal MDS and non-metrc MDS can be employed n ths work. Classcal MDS preserves the dssmlarty dstances better. However, snce our am s the estmaton of the quantles of producton, non-metrc MDS, where only ranks are mportant, not absolute dstances, s also vald. Prelmnary tests usng these two methods have not shown sgnfcant dfferences n the results. Note that snce the map obtaned by MDS s derved solely by the dssmlarty dstances n the matrx, the absolute locaton of the ponts s rrelevant. The map can be subject to translaton, rotaton, and reflecton, wthout mpact to the methodology. Only the dstances n mappng space R are of nterest. For more detals about MDS methodologes, see Appendx A and Borg and Groenen (997). 5

6 .. Applcaton of MDS To understand better the concept of MDS, we reconsder the example of the three ctes n US: New York, San Francsco and Palo Alto. Ther dstances are presented n Table. SF Palo Alto N-Y SF Palo Alto N-Y Table : Dstances between ctes Gven a matrx of dstance between ctes, MDS plots the ctes n a map such that the dstances are respected (assumng classcal MDS has been appled). Ctes are mapped n such a way that ctes wth small dstances to each other are placed near each other on the map, and those ctes that have large dstances between each other are placed far away on the map. Note that ths map s derved only from dstances between ponts, thus, the orentaton of the map may not correspond to the standard map wth North-South and East-West axes wthout further rotaton, translaton, and reflecton. But ths has no mportance snce we do not need to know where the ponts are exactly located n the map - only ther Eucldean dstances are requred. Applyng ths concept to our methodology, the objects under consderaton are reservor models, thus each reservor model s represented as a pont. MDS allows to represent N each realzaton y R C, =,, N (defned by potentally mllons of grd-blocks) n a reduced coordnate system x 3 to vsualze uncertanty). R p R, =,, N R, where p s usually small (p = or An llustraton of an applcaton of classcal MDS s presented n Fgure, llustrated for faces models and Hausdorff dstance as a dssmlarty matrx (Suzuk and Caers, 6). A D Eucldean space s suffcent to have a good mappng, n other words the Eucldean dstances between any two ponts are very close to the dstances n the dssmlarty matrx. Ths space allows the vsualzaton of uncertanty. Assumng that the dstance s well correlated to the producton response, ponts close to each other have smlar responses (and smlar geologcal structures). 6

7 Fgure : Multdmensonal Scalng (MDS): each pont represents a reservor model n D space. For most applcatons, the structure of the ponts n the mappng space R s not lnear and therefore standard tools for pattern detecton are not approprate. To avod ths problem, the ntroducton of kernel methods, partcularly kernel prncpal component analyss wll be very useful..3 Kernel Prncpal Component Analyss Kernel prncpal component analyss (KPCA) was ntroduced by Schölkopf (996) as a nonlnear generalzaton of prncpal component analyss. Ths theory was recently developed from the feld of neural computng and pattern recognton, and s often used as a tool to remove nose from computerzed mages. In the petroleum feld, KPCA has been used by Sarma (6) n the context of hstory matchng and producton optmzaton. In our applcaton, we use kernel methods to transform ponts generated by MDS n nonlnear space R nto a space F wth mproved lnear varaton. Once n ths lnear space, standard tools such as prncpal component analyss and cluster analyss are employed to analyze the pont structure. Fgure 3 llustrates the mappng from space R to space F, and a subsequent determnaton of the prncpal component vector (shown n red). 7

8 R F Φ Fgure 3: Kernel Prncpal Component Analyss: Prncpal Component Analyss n a transformed space F, whch has mproved lnear varaton (modfed from Schölkopf et al, ).3. Summary of KPCA To understand KPCA better, we frst recall quckly the theory of prncpal component analyss (PCA). PCA conssts of projectng the data onto a lower-dmensonal lnear space, whch account for most of the structure n the data. PCA provdes a set of orthogonal axes, called prncpal components, obtaned by solvng the egenvalue problem of the sample correlaton matrx. A small number of prncpal components s often suffcent to descrbe the major trend n the data. KPCA works n a smlar manner: frst, kernels map the gven data ponts from space R to space F usng a multdmensonal functon Φ : Φ : R F and then PCA s appled n F. It can be shown that by combnng kernels and PCA n a sngle procedure, there s no need to map explctly the ponts from space R to F: all necessary computatons n space F can be carred out usng the nonlnear functon Φ n nput space R. Ths functon s called a kernel functon k, and s gven by: k ( x, y) = Φ( x), Φ( y) () Thus, even f the space F and the mappng Φ are complcated, KPCA s formulated n such a way that only the dot product n F s needed (Eq. ). A commonly used kernel functon s the Gaussan kernel (radal bass functon), gven by: x y k ( x, y) = exp wth σ > () σ It can be shown that KPCA requres only the egenvectors and egenvalues of the N R x N R kernel matrx K defned by: K = k ( x, x ), x R, =,, N j j Applyng KPCA to a dataset requres the defnton of a kernel functon, ts parameters, and the number of egenvalues of the matrx K to retan. In our applcaton, we consder a Gaussan kernel for all the cases (Eq. ). The parameter σ controls the flexblty of the R 8

9 kernel. For small values of σ, the kernel matrx becomes close to dentty matrx (K=I) and thus the egenvalues are on the same order of magntude. On the other hand, large values of σ gradually reduce the kernel to a constant functon (K=). In ths case, only a small number of egenvalues are sgnfcant. Thus, a compromse must be made when selectng σ. The parameter σ and the number of retaned egenvalues must be chosen n order to have a balance between the qualty of PCA and the egenvalue structure of the kernel matrx. The objectve s to select σ such that when retanng only a small number of large egenvalues for K we obtan a PCA of good qualty (as measured by the percentage of explaned varance). In the applcatons below, we select σ such that the error made by projectng the ponts nto the lower-dmensonal space (3 to 5D) of the feature space s less than 5%. The mathematcal detals of KPCA are descrbed n Appendx B. Below, we llustrate the method usng a smple example. For ths example, we need to frst ntroduce the concept of the pre-mage problem..3. Pre-mage constructon Whle the mappng Φ from nput space R to feature space F s of prmary mportance n kernel methods, the reverse mappng from feature space F back to nput space R may be desred (Fgure e). Ths reverse mappng process s called the pre-mage problem. For example, one may want to map back n R the ponts projected by KPCA nto the lower dmensonal space F. The dffculty n ths procedure s that the mappng functon Φ nto the feature space s not known, nonlnear and non-unque, thus only approxmate solutons are possble. We denote by z the approxmate pre-mage of Ψ f ρ ( z) = Ψ Φ( z) s mnmal. In ths work, we used the fxed-pont teraton approach proposed by Schölkopf, n order to fnd approxmate pre-mages. Ths approach s essentally a gradent-based optmzaton technque. In the llustratve examples below, the pre-mage ponts are shown n red. Note that the red pre-mage ponts do not necessarly correspond to a blue pont n the orgnal space R. In ths case, we take the closest exstng pont. For detals about the pre-mages algorthm, see Schölkopf et al. ()..3.3 Illustratve Example usng PCA and KPCA We consder n ths part, a smple example taken from Schölkopf et al. (). An artfcal data set s generated from three Gaussans models (standard devaton σ =.), each one contanng 5 ponts. Fgure 4 shows the pont representaton. 9

10 Fgure 4: Two-dmensonal example wth three data clusters We frst perform lnear PCA (Fgure 5) by projectng the ponts on the frst prncpal component. Note that PCA cannot dentfy the three clusters, due to the non-lnearty of the pont structure. In ths case, the projectons of the ponts are separated nto only groups, one above the red lne and one below t Fgure 5: Prncpal Component Analyss In Fgure 6, we perform KPCA wth a Gaussan kernel (σ =.), projectng the data usng from to 6 prncpal components. The ponts were then reconstructed n the ntal space, usng the Schölkopf fx-pont algorthm. Fgure 6 shows that dscardng hgherorder prncpal components s analogous to a removal of nose around the center of each cluster. Ths smple example clearly shows a nonlnear approxmaton of cluster behavor, even when only one prncpal component s retaned. When more prncpal components are retaned, the varaton wthn the clusters centers s ncreasngly better and nonlnearly approxmated.

11 Fgure 6: Kernel Prncpal Component Analyss: Blue ponts are orgnally ponts. Red dots are backtransformed KPCA mappngs from space F.3.4 KPCA Fnal Remarks KPCA s a powerful technque for extractng structure from potentally hgh-dmensonal data sets wth complex varablty. KPCA can be seen as a way of nferrng a lowdmensonal subspace of F that mantans the prncpal characterstcs of the ponts n R and thus can be consdered as removng nose from the ponts n R. Note that the dmenson of the subspace of F may be larger than the dmenson of R. In our applcaton, we apply KPCA to hundreds of realzatons mapped n a Eucldean space. The subspace of F dentfes a small subset of realzatons whch represent typcal realzatons of the full set. Full flow smulatons are then performed on ths subset of realzatons. Uncertanty can be subsequently analyzed by calculatng, for example, the quantles P, P5 and P9 on these few models as a functon of smulaton tme. We present an applcaton of KPCA usng a synthetc reservor case n Secton 3..4 Kernel K-Means Clusterng (KKM) Snce the kernel method transforms the data ponts nto a lnear space F, we can apply technques other than PCA n space F. Clusterng algorthms are also applcable and are suted to our problem. Cluster analyss ams to dscover the nternal organzaton of a

12 dataset by fndng structure wthn the data n the form of clusters. Hence, the data s broken down nto a number of groups composed of smlar objects. Ths methodology s wdely used both n multvarate statstcal analyss and n machne learnng. Defnng clusters conssts n dentfyng an a pror fxed number of centers and assgn ponts to cluster wth the closest center. In ths work, we apply the classcal k-means algorthm n feature space F to determne a subset of ponts defned by the cluster centrods. The k-means algorthm tres to assgn ponts n k clusters S by mnmzng the expected squared dstance between the ponts of the cluster and ts center µ : J = k = x S j x µ The algorthm starts by parttonng randomly the nput ponts nto k ntal sets S. It then calculates the mean pont, or centrod µ, of each set. Then, every pont s assgned to the cluster whose centrod s closest to that pont. These two steps are alternated untl convergence, whch s obtaned when the ponts no longer swtch clusters (or alternatvely centrods are no longer changed). The k-means procedure requres a method for measurng the dstance between two ponts n the hgh-dmensonal feature space F. Once agan, ths Eucldean dstance can always be computed usng the nner product nformaton through the equalty: Φ ( x) Φ( z) = Φ( x), Φ( x) + j Φ( z), Φ( z) = k( x, x) + k( z, z) k( x, z) Φ( x), Φ( z) Note that ths equalty s only true for Eucldean dstance, hence the necessty of the MDS procedure pror to performng KKM. For an overvew of clusterng technques, see Buhmann (995), and Shawe-Taylor and Crstann (4) for specfc nformaton about kernel clusterng technques. Prelmnary results of the applcaton of ths methodology are presented n Secton 3. 3 Applcaton to a synthetc reservor A synthetc case study s presented to demonstrate the potental of the proposed methodology. We consder a channel system, composed of mud and sand. The channel sands have unform porosty and permeablty, the mud s treated as nactve cells. The reservor model s a 8x8 D grd contanng 3 producers and 3 njectors, all penetratng channel sand. To vew the locaton of the wells and examples of the faces models, see Fgure 7. 8 tranng mages of possble geologcal scenaros were used to generate 45 faces models. In order to analyze the effcency and qualty of the method, Eclpse

13 smulatons were run for each model. Note that for real feld cases, ths s n general not possble. For a more detaled descrpton of the case, see Suzuk and Caers 6. Sand Shale Fgure 7: Example of 3 reservor models wth well locatons We now apply the methodology proposed n ths paper to ths case usng dfferent dstance measures based upon streamlne smulaton. Streamlne smulaton has been shown to be orders of magntude faster than standard flow smulaton, and s thus well suted for problems where rapd evaluaton of many models s needed. We also compare our results wth tradtonal methods, such as rankng wth statc propertes and tracer smulatons. 3. Applcaton of the proposed methodology Case In ths secton, we present the results followng the methodology as shown n Fgure. 3.. Constructon of the dstance matrx The dstance s calculated usng a streamlne smulator (3DSL). The streamlne smulaton approxmates the fluds as ncompressble, and smulaton was performed for only two tme steps up to the end of producton (at 7 days). The dstance between any two reservor models s gven as the absolute dfference n total feld ol rate () at 7 days: streamlne streamlne δ = (7) (7) (3) j j As dscussed before, the dstance needs to be reasonably well correlated wth the flow response we are nterested n. To compare the dstance wth the results usng Eclpse, we calculate the average of absolute dfference n ol rate as: Nt Eclpse Eclpse j = ( t) j ( t) (4) Nt t= where t represent the tme and N t the number of tmesteps. 3

14 Fgure 8a plots the madogram,.e. the dstance δ j versus the average absolute dfference n ol rate j. Note that for ths synthetc case, the dstance s well correlated to the dfference n ol rate (correlaton coeffcent ρ ( δ, ) =. 84 ). Fgure 8b plots the correspondng varogram,.e. the squared dfference n ol rate versus the dstance δ j. ρ =.84 (a) Fgure 8: Ol rate as a functon of the dstance: (a) Madogram, (b) Varogram (b) 3.. Mult-Dmensonal Scalng Usng the dssmlarty dstance prevously defned, we appled classcal multdmensonal scalng to map all the realzatons n a -dmensonal Eucldean space R (Fgure 9a). Fgure 9b represents the Eucldean dstances between any two ponts n D space R vs. the dssmlartes between any two realzatons. The Eucldean dstances between ponts n R reproduce almost exactly the dssmlarty dstances. The correlaton coeffcent between the dssmlarty matrx D and the par-wse Eucldean dstance s Eucldean Dstance n D (a) R Dssmlarty Dstance Fgure 9: MDS (a) D Eucldean Space of Uncertanty (b) Correlaton between dssmlarty matrx and Eucldean dstance (b) 4

15 Fgure 9b demonstrates that Eucldean dstance s a very good representaton of the dssmlartes of the reservor models. From ths pont, we only consder these Eucldean dstances between the models Applcaton of KPCA At ths step of the methodology, we defne a kernel functon whch allows transformng the mappng space R nto a space F wth mproved lnear varaton. The Gaussan radal bass functon kernel s used (Eq. ). We adjust the kernel parameters (parameter σ and number of retaned egenvectors) n order to remove the nose from the scatter of ponts and thus to obtan relatvely few realzatons. For ths case, we perform KPCA wth σ = 45 and retan only prncpal component. We then compute the pre-mages of each pont, usng Schölkopf fxed-pont algorthm, wth the pre-mages presented n blue. After undergong KPCA, the 45 ponts are mapped to only 5 dstnct locatons, blue ponts can be nterpreted as a denosed verson of red ponts..8 Input Reconstruted Fgure : Mappng space R: Blue ponts represent the ponts selected by KPCA R Full flow smulatons are performed for 5 realzatons correspondng to the blue ponts. Recall that the pre-mage mappng may not result n ponts correspondng to a realzaton. In ths nstance, we select the nearest pont (realzaton) for flow smulaton. Fgure represents the feld ol rate as a functon of tme for all 45 realzatons (Fgure a) and for the 5 realzatons (Fgure b). 5

16 (a) Tme (days) Fgure : Feld ol rate as a functon of tme (a) for the 5 selected realzatons, (b) for all 45 realzatons, (c) resultng quantles We can see that ponts selected are well spaced (Fgure ) and gve a good overall spread of response (Fgure b). However, the resultng quantle estmaton s not very accurate (Fgure c). Ths can be explaned by the fact that each of the 5 selected realzatons does not have the same frequency n the entre set - medum values of feld ol rate are proportonally overrepresented compared to hgh and low values (Fgure b). Thus, a weghtng scheme should be defned for a proper estmaton of the quantles, where each realzaton should be represented as many tmes as ts frequency n the 45 realzatons. It s mportant to note that snce the mappng space R s nonlnear, one cannot use smple nearest-neghbor counts or clusterng technques n ths space R to obtan these weghts. To defne a weghtng scheme, we apply a clusterng algorthm (k-means) n the new denosed subspace of F constructed usng KPCA. Cluster analyss allows the selecton of a small (user-defned) number of representatve ponts (centrods of the clusters) on whch we wll perform flow smulatons. The weghtng scheme s then easy to compute, each model beng represented as many tmes as the number of models n the correspondng cluster. We use n ths applcaton the common Gaussan radal bass functon kernel (Eq. ) usngσ = 5. We frst perform KPCA to reduce the dmensonalty of the problem, by projectng the ponts n a 3D subspace of the feature space, whch we denote F (Fgure a). In the subspace F, we perform cluster analyss usng k-means, to determne 5 clusters. The number of cluster was defned as the maxmum number of flow smulatons we can afford for a gven CPU. The centrods of the 5 clusters defne typcal models for whch flow smulaton wll be performed. These centrods are presented n blue n Fgure b. (b) (c) 6

17 models models for smulaton (a) -.5 F Fgure : (a) 3D subset of F, (b) Mappng space R: Blue ponts represent the ponts selected by KPCA and k-means (b) R Uncertanty quantfcaton s then performed by calculatng the quantles P, P5 and P9 on these 5 models as a functon of tme, each model beng represented as many tmes as the number of models n the correspondng cluster. In ths case, only 5 flow smulatons were performed, for a total of 45 reservor models. Results for feld ol rate are presented n Fgure 3. Fgure 3a represents the evoluton of ol rate as a functon of tme for the 5 selected realzatons and Fgure 3b represents the resultng quantles, n blue and the quantles resultng from the 45 smulatons Tme (days) (a) All realzatons KPCA realzatons Fgure 3: Ol Rate as a functon of tme (a) for all the 5 realzatons, (b) P P5 P9 values (b) As we can observe on Fgure 3, the estmaton of quantles P, P5 and P9 s accurate. The red curves and blue curves are very smlar. In Fgure 4, we present the probablty densty of the feld ol rate for the 45 realzatons (n red), and for the weghed 5 realzatons (n blue) for 3 dfferent tmes (, 4 and 6 days). We can see that the estmated denstes are close to the reference, whch means that the 5 realzatons have smlar characterstcs as the 45. 7

18 . x -3 All Data Kpca results Densty Plot at days 4.5 x All Data Kpca results Densty Plot at 6 days 4.5 x All Data Kpca results Densty Plot at 4 days Frequency.6 Frequency.5 Frequency (a) Fgure 4: Densty of ol rate for all 45 realzatons (red) and 5 selected realzatons (blue) (a) days, (b) 4 days, (c) 6 days (b) (c) In order to demonstrate the effectveness of the methodology, we compare the results wth those obtaned by choosng randomly 5 realzatons. To do so, we have also estmated the quantles for feld ol rate for a random selecton of 5 realzatons where smulatons are performed. Fgure 5b presents the best set of quantles and n Fgure 5c the worst set of quantles obtaned for dfferent sets of 5 random models. All realzatons KPCA realzatons All realzatons Random realzatons All realzatons Random realzatons (a) (b) (c) Fgure 5: P P5 P9 values as a functon of tme, (a) proposed method, (b) best results usng 5 randomly chosen models, (c) worst results usng 5 randomly chosen models Fgure 5 shows that selectng realzatons for smulaton processng usng a combnaton of KPCA and k-means s much more accurate than usng 5 randomly chosen models, even for the most fortutous random selecton. We now propose a second procedure whch replaces ths two-step methodology (PCA and clusterng) by performng a sngle cluster analyss n the feature space F Applcaton of KKM In ths secton, we apply the second methodology, kernel k-means (KKM), to the same case. The two frst steps, the defnton of the dssmlarty matrx and mappng the ponts wth MDS, are dentcal for the methods, thus they are not llustrated here. 8

19 A Gaussan kernel (Eq. ) wth σ = 65 s used to defne the feature space F n whch clusters are dentfed. Agan, we suppose that only 5 flow smulatons are affordable n ths case. Fgure 6 represents all 45 realzatons n the mappng space, the blue ponts are the centrods of 5 clusters detected n F All realzatons Selected realzatons Fgure 6: Mappng space R: Blue ponts represent the ponts selected by kernel k-means R Uncertanty quantfcaton s subsequently performed by flow smulaton for the 5 realzatons selected by KKM and by computng the resultng quantles P, P5 and P9. As mentoned before, the weght of each realzaton equals the number of ponts n the correspondng clusters. Results are shown on Fgure Tme (days) (a) 5 All realzatons KKM KPCA realzatons Tme (days) Fgure 7: Ol Rate as a functon of tme (a) for all the 5 realzatons, (b) P P5 P9 values (b) Fgure 8 represents the densty computed from the 5 smulatons (n blue), as well as the densty for the full set of realzatons for 3 dfferent tmes. We can see that the subset of 5 selected realzatons has smlar probablty denstes compared the entre set of 45 realzatons. 9

20 .9.8 x -3 Densty Plot at days 4.5 x -4 Densty Plot at 6 days 6 x -4 Densty Plot at 4 days All Data All Data All Data KKM Kpca results KKM Kpca results 4 KKM Kpca results Frequency Frequency 4 3 Frequency (a) (b) Fgure 8: Densty of ol rate for all 45 realzatons (red) and 5 selected realzatons (blue) (a) days, (b) 4 days, (c) 6 days (c) We now consder a second example, usng a second dstance measure whch s less correlated wth the flow response than n the prevous case. 3. Applcaton of the proposed methodology Case 3.. Constructon of the dstance matrx Once agan, the dstance s calculated usng a streamlne flow smulator (3DSL). However, n ths case we perform tracer smulatons. A tracer smulaton approxmates the flow as lnear, meanng that the njecton and producton fluds are assumed to be dentcal. Thus, only a sngle pressure solve s necessary to perform flud flow smulaton, gvng extremely rapd results. The dstance between any two reservor models s gven as the absolute dfference n feld ol rate at two gven tmes ( and days). δ = j t {,} streamlne ( t) streamlne j Fgure 9a represents the madogram of the absolute dfference n flow (Eq. 4) as a functon of the dstance δ, as well as the correspondng correlaton coeffcent. Fgure 9b represents the varogram,.e. the squared dfference n ol rate versus the dstance δ j. ( t)

21 ρ =.77 (a) (b) Fgure 9: Ol rate as a functon of dstance (a) Madogram, (b) Varogram For ths synthetc case, the dstance s reasonably correlated to the absolute dfference n flow response (correlaton coeffcent: ρ ( δ, ) =. 77 ). The correlaton s weaker than n Case above, due to the larger approxmatons n modelng the flow usng tracer streamlne smulaton. 3.. Mult-Dmensonal Scalng In ths example, a 3D mappng space R s deemed approprate to ensure that the Eucldean dstance between any two ponts n R reproduces the dssmlarty n the matrx D (Fgure a). Indeed, the correlaton coeffcent between the dssmlarty matrx D and the par-wse Eucldean dstance s hgh at.9 (cf. fgure). The 3D mappng space R representng all the 45 realzatons s presented on Fgure b Eucldean Dstance n 3D Dssmlarty Dstance (a) R - Fgure : MDS: (a) Correlaton between dssmlarty matrx and Eucldean dstance, (b) 3D Eucldean Space of Uncertanty (b)

22 3..3 Applcaton of KPCA We use a Gaussan radal bass functon kernel (Eq. ) wth σ = 8. We frst perform KPCA to reduce the dmensonalty of the problem, by projectng the ponts n a 4D subspace of the feature space. In that subspace, we perform cluster analyss usng k- means, to determne 5 clusters. The centrods of the 5 clusters defne representatve models for whch flow smulaton wll be performed. Uncertanty quantfcaton s then performed on these 5 models, each model beng represented as many tmes as the number of models n the correspondng cluster. In ths case, 5 flow smulatons were performed, for a total of 45 reservor models. Results for feld ol rate are presented n Fgure Tme (days) All realzatons KPCA realzatons (a) (b) Fgure : Ol Rate as a functon of tme (a) for all the 5 realzatons, (b) P P5 P9 values In Fgure a, we represented the ol rate as a functon of the tme correspondng to all 5 realzatons where full flow smulatons where performed. The resultng quantles of the methodology (Fgure b) are accurate. Denstes of the selected realzatons and the full set are presented n Fgure. Agan, the estmated denstes are of good qualty. 8 x -4 7 All Data Kpca results Densty Plot at days 4.5 x -4 4 All Data Kpca results Densty Plot at 4 days 4.5 x -4 4 All Data Kpca results Densty Plot at 6 days Frequency Frequency Frequency (a) (b) Fgure : Densty of ol rate for all 45 realzatons (red) and 5 selected realzatons (blue) (a) days, (b) 4 days, (c) 6 days (c)

23 In ths case, although the correlaton between the dstance and the flow response s weaker compared to the frst case, results usng the new approach are much more accurate than those obtaned by selectng 5 randomly chosen realzatons (refer to Fgures 5b and 5c) Applcaton of KKM We now apply the KKM algorthm to calculate the quantles P, P5 and P9 of the producton. In ths case, we represent the ponts n a D mappng space R. A Gaussan kernel was used, whose parameter s σ = 85. Agan, we choose to create 5 clusters, n order to perform only 5 full flow smulatons. Quantles resultng form the 5 selected ponts are presented n Fgure 3b, probablty denstes of the feld ol rate for dfferent tme steps are presented n Fgure Tme (days) (a) All realzatons KKM KPCA realzatons Fgure 3: Ol Rate as a functon of tme (a) for all the 5 realzatons, (b) P P5 P9 values (b).9.8 x -3 Densty Plot at days 6 x -4 Densty Plot at 4 days 6 x -4 Densty Plot at 6 days All Data All Data All Data Kpca KKM results KKM Kpca results Kpca KKM results 5 5 Frequency Frequency 4 3 Frequency (a) (b) Fgure 4: Densty of ol rate for all 45 realzatons (red) and 5 selected realzatons (blue) (a) days, (b) 4 days, (c) 6 days Uncertanty analyss for the 5 selected realzatons s agan of good qualty, the resultng quantles and denstes are close to the reference (n red). (c) 3

24 3.3 Comparson of the methodology wth classcal rankng technques In ths secton, we propose to compare the results obtaned by the proposed methodology wth the classcal technque whch conssts of rankng the realzatons accordng to a specfc measure and then determne realzatons for flow smulaton processng Rankng technque revew The dea of rankng stochastc realzatons was frst publshed n the context of stochastc reservor modelng n 99 (Balln et al.). The central dea behnd rankng realzatons s to use some smpler measure to rank realzatons and then run full flow smulaton wth fewer realzatons, for example for those whch represent P, P5 and P9. Ths would defne the bounds of the uncertanty wthout performng a large number of fne-scale flow smulatons. The central goal of rankng s to explot a relatvely smple statc measure to accurately select geologcal realzatons that correspond to the targeted percentles of the producton responses. The rankng and selectng of realzatons must be talored to the flow process. It s well known that a partcular rankng measure must be hghly correlated to producton response. Conventonal rankng measures are, for example, orgnal ol n place or connectvty (McLennan and Deutsch, 5). Streamlne smulaton (Glman et al. ) and tracer smulaton (Balln et al., 99) have receved sgnfcant attenton n the past few years. However, there s no unque rankng ndex when there are multple flow response varables and no rankng measure s perfect Comparson of quantle estmaton In ths work, we have consdered two dfferent measures for each of the 45 realzatons: orgnal ol n place (OOIP) and ol rate obtaned by streamlne tracer smulaton, as used for the dssmlarty dstance n case. Once a rankng measure s selected, the methodology for rankng and selectng geostatstcal realzatons for flow processng s straghtforward. The rankng measure s calculated for every geostatstcal realzaton. The low (P), medum (P5) and hgh (P9) geologcal realzatons are then selected for flow modelng. Results for ol rate are presented n Fgure 5. We have presented n red the quantles obtaned wth the entre set of realzatons, and n blue the quantles resultng from the rankng measure. Fgure 5a represents results usng OOIP as a rankng measure, Fgure 5b to Fgure 5d represent results usng the ol rate from the tracer at respectvely 7, and days. 4

25 All realzatons OOIP KPCA realzatons All realzatons Tracer KPCA realzatons 7 d (a) (b) All realzatons Tracer KPCA realzatons d All realzatons Tracer KPCA realzatons d (c) Fgure 5: Quantles P, P5 and P9 resultng from rankng measures- (a) OOIP, (b) Tracer - 7 days, (c) Tracer - days, (d) Tracer - days. As we can see on Fgure 5, the quantles estmatons usng rankng method wth OOIP and streamlne tracer are less accurate than the one obtaned wth the proposed methodology. To understand why the OOIP and tracer rankngs provde less accurate results, we plot n Fgure 6 the feld ol rate (from Eclpse) for all 45 models as a functon of the rankng measurement. In order for the rankng procedure to gve relable results, the rankng measure must be strongly correlated wth the flow producton. Fgure 6 shows that OOIP may not be a good measure for rankng realzatons (correlaton coeffcent between OOIP and feld ol rate s.9). Here, tracer smulaton s more sutable due to the mproved correlaton wth the flow response (.63,.76 and.87 respectvely). (d) 5

26 . x 5 ρ =.9. x 5 ρ = OOIP x. x ρ =.76 (a) Ol Rate - tracer - 7 days. x ρ =.87 (b) Ol Rate - tracer - days (c) Ol Rate - tracer - days Fgure 6: Ol rate producton as a functon of: (a) OOIP, (b) Tracer 7 days, (c) Tracer days, (d) Tracer days (d) The rankng methods for selectng the P, P5 and P9 realzatons for flow smulaton are not as accurate as the methodology proposed n ths paper. However, n Fgure 5, only 3 full flow realzatons were performed, whereas for the new method, 5 flow smulatons were necessary. To compare both methods based upon the same number of flow smulatons, we propose to select 5 realzatons equally spaced accordng to the rankng measure. Resultng quantles for feld ol rate are presented n Fgure 7. 6

27 All realzatons OOIP KPCA realzatons All realzatons Tracer KPCA realzatons 7 d (a) (b) All realzatons KPCA Tracer realzatons d All realzatons Tracer KPCA realzatons d (c) Fgure 7: Quantles P, P5 and P9 resultng from rankng measures- (a) OOIP, (b) Tracer - 7 days, (c) Tracer - days, (d) Tracer - days For the same number of flow smulaton, Fgure 7 shows that the use of rankng measures s less accurate than the use of KPCA or KKM. In addton, we observed n our work that the quantle estmaton s hghly dependent on whch 5 realzatons are selected. Note that n the method proposed n ths paper, we use tracer smulatons for calculatng the dstance but not for rankng - the selecton of realzatons s done usng another approach (KPCA or KKM). In the case shown here, results show that for the same measure (tracer smulaton), better results are obtaned from KPCA or KKM than from rankng. Ths s llustrated n Fgure 8. (d) 7

28 ρ Absolute values for rankng: Tracer Eclpse (, ) =. 87 Rankng Tracer Smulaton and days Dstance for kernel method: Tracer Eclpse ρ, =. ( ) 77 KPCA Fgure 8: Comparson between rankng and KPCA usng the same tracer measure. Note that the correlaton coeffcent for the absolute values of rankng measure s greater than the relatve (dstance) measure, but the P, P5, P9 estmatons are less accurate. In Fgure 8, we observe that the effcency of the rankng technque reles on a hgh correlaton coeffcent between the rankng measurement and the flow response, whereas n case of dstances, a smaller correlaton coeffcent s suffcent. Also, for an equvalent correlaton coeffcent, quantles are more accurate usng dstances and kernels, than usng rankng measures (Fgure 9). ρ Absolute Values for rankng Tracer: days Tracer Eclpse (, ) =. 76 Rankng ρ Dstance for kernel method Tracer: and days Tracer Eclpse (, ) =. 77 KPCA Fgure 9: Comparson between rankng and KPCA for smlar correlaton coeffcent. Note that for smlar correlaton coeffcent, P, P5, P9 estmatons are more accurate for kernels. 8

29 4 Conclusons and future work We have presented prelmnary results for a new model-based method for uncertanty quantfcaton. The method reles on a reasonable correlaton between the dstance measure and the producton responses of nterest. In our example, we use streamlne smulaton to obtan the dstances, whch correlated well wth the dfferences n producton response usng standard flow smulaton. Gven the dstance measure, we employ KPCA, and k-means clusterng or kernel k-means to select a subset of 5 reservor models whch contan the same P, P5, P9 quantles as for the entre set of 45 models. The applcaton of ths new method shows promsng results - quantle estmatons usng ths methodology are notceably better than those usng tradtonal rankng methods. Future work ncludes fndng ways to better optmze the Gaussan kernel parameter value and lookng at alternatve kernel functons. Other dstance measurements should also be tested, as well as more complex synthetc cases. Comparson wth the tradtonal method of expermental desgn would be also of great nterest. Acknowledgments The authors would lke to acknowledge SCRF sponsors and Chevron for ther support. Many thanks as well to Darryl Fenwck from StreamSm Technologes for hs help usng 3DSL, and for our many useful dscussons. References Arpat, B. G. [5] Sequental Smulaton wth patterns, Ph. D dssertaton, Stanford Unversty Balln, P.R., Journel A.G., and Azz, K. [99] Predcton of Uncertanty n Reservor Performance Forecast, JCPT, no. 4 Borg, I., Groenen, P. [997] Modern multdmensonal scalng: theory and applcatons. New-York, Sprnger Buhmann, J. M. [995] Data clusterng and learnng. The Handbook of Bran Theory and Neural Networks, pages 78-8, MIT Press. Glman, J.R., Meng, H.-Z., Uland, M. J., Dzurman, P.J., Cosc, S. [] Statstcal Rankng of Stochastc Geomodels Usng Streamlne Smulaton: A Feld Applcaton. SPE Annual Techncal Conference and Exhbton, SPE McLennan, J.A., and Deutsch, C.V. [5] Rankng Geostatstcal Realzatons by Measures of Connectvty, SPE/PS-CIM/CHOA. Sarma, P., [6]. Effcent Closed-Loop Optmal Control of Petroleum Reservors Under Uncertanty. PhD Dssertaton, Stanford Unversty, USA. 9

30 Shawe-Taylor John, Crstann Nello, [4] Kernel Methods for Pattern Analyss, Cambrdge Unversty Press Schöelkopf, B., Smola, A. J., Muller, K.-R. [996] Nonlnear component analyss as a kernel egenvalue problem, Techncal Report 44, Max-Planck-Insttut für bologsche Kybernetk. Schöelkopf, B., Smola, A. [] Learnng wth Kernels, MIT Press, Cambrdge, MA. Suzuk, S., Caers, J. [6] Hstory matchng wth an uncertan geologcal scenaro. SPE Annual Techncal Conference and Exhbton, SPE 54. 3

31 Appendx A: Mult-Dmensonal Scalng (MDS) We present n ths appendx the man steps of the two basc types of MDS: classcal MDS and non-metrc MDS. The startng pont of each MDS s a matrx consstng of the parwse dssmlartes of the objects n consderaton D = δ j, where δ j s the dssmlarty between objects and j. Classcal MDS Let the coordnates of N R ponts n p dmensons (models n our applcaton) be denoted x, =,, N R. These can be collected together n the N R xp matrx: X = [x, x,, x NR ]. The classcal MDS algorthm rests on the fact that the coordnate matrx X can be derved by egenvalue decomposton from a Gram matrx B, whch s obtaned by convertng the dssmlarty matrx D nto a scalar product. The followng steps summarze the algorthm of classcal MDS:. Construct a matrx A wth elements aj = δ j T. Construct a matrx B by centerng A: B = HAH usng the matrx H = I n 3. Extract the p largest postve egenvalues λ λ p of B and the correspondng p egenvectors e e p. 4. A p-dmensonal spatal confguraton of the N R objects s derved from the coordnate matrx / X = E p p where p s the dagonal matrx of p egenvalues of B, respectvely. E s the matrx of p egenvectors and Classcal MDS assumes the dstances to be Eucldean. However, n many applcatons, the data are not dstances as measured from a map, but rather dssmlartes data. When applyng classcal MDS to dssmlartes, t s assumed that the proxmtes behave lke real measured dstances. The advantage of classcal MDS s that t provdes an analytcal soluton, requrng no teratve procedures. Non-metrc MDS The dea of nonmetrc MDS s to demand a less rgd relatonshp between the dssmlartes and the dstances. Only the ordnal nformaton n the dssmlartes s used for constructng the spatal confguraton. p 3

32 The gven dssmlartes δ j are used to generate a set of dstances d j (also called dspartes), whch are approxmately related to the gven dssmlartes δ j by a d f δ. The functon f has the property: monotonc ncreasng functon f such as ( ) j δ j < δ rs f ( δ j ) < f ( δ rs ) The most common approach to determne the elements d j and the underlyng confguraton x, x,, x NR s an teratve process, commonly referred to as the Sheppard- Kruskal algorthm. The fttng s obtaned by mnmzng the so-called STRESS,.e., the sum of squares of the dfferences between the dstances and the correspondng pseudodstances over all the ponts of the confguraton STRESS =, j j ( f (δ j ) dj ) d, j j The core of a nonmetrc MDS algorthm s a twofold optmzaton process. Frst the optmal monotonc transformaton of the dssmlartes has to be found. Secondly, the ponts of a confguraton have to be optmally arranged, so that ther dstances match the scaled dssmlartes as closed as possble. The bascs steps of non-metrc MDS are:. Assgn ponts to arbtrary coordnates n p-dmensonal space: X = [x,x,,xn R ].. Compute Eucldean dstances among all pars of ponts: d j = d, T ( x x ) = ( x x ) ( x x ) j j j 3. Evaluate the STRESS functon: the smaller the value, the greater the correspondence 4. Adjust coordnates of each pont n the drecton that best maxmze the STRESS 5. Repeat steps through 4 untl STRESS don t get any lower. Remarks: To be effectve, the dssmlarty matrx D should be postve defnte. However, mappng can be done accurately by only consderng postves egenvalues, f negatve egenvalues are of small ampltude. 3

33 Appendx B: Kernel Prncpal Component Analyss (KPCA) We descrbe, n ths appendx, the KPCA algorthm n more detal. Gven a set ponts representng reservor models x, =,, N, x R and a multdmensonal functon Φ : R F, x Φ(x) the covarance C n the feature space F s gven by: N R j= R N R T C = Φ( x ) Φ( x ) () We now have to fnd the egenvalues > and non-zeros egenvectors v n F such as: v = Cv () As the feature space F could have an arbtrarly large, and possble nfnte dmenson, solvng ths problem wth standard algorthm s a very expensve process. However, as dscussed n Schölkopf (996), an alternatve but exactly equvalent formulaton of the same problem can be solve more effcently to determne the non-zeros egenvalues and the egenvectors v of covarance matrx C. Ths alternatve s based on the fact that all solutons v wth le n the span of Φ x ), Φ( x ),, Φ( ). Ths has two useful consequences. ( x N R j j Frst, we may also consder the set of equatons: Φ ( xn ), v = Φ( xn ), Cv for all n =,, N R (3) Second there exst coeffcents α, =,, N such that: v NR = = Combnng these consequences together, we get: N R = R R α Φ( x ) (4) N N α Φ( x n ), Φ( x ) = α Φ( x ), Φ( x j ) Φ( x j ), Φ( x ) for all n =,, N R R n N R = j= Whch s equvalent to: N RK α = K α where K s the N R xn R matrx such as: T K = Φ( x ) Φ( x (5) j j ) Fnally, n order to solve the egenvalue problem (), we only need to solve the egenvalue problem: N R α = Kα (6) 33