On incorporating time-lapse seismic survey data into automatic history matching of reservoir simulations

Transcription

1 ontents On ncorporatng tme-lapse sesmc survey data On ncorporatng tme-lapse sesmc survey data nto automatc hstory matchng of reservor smulatons Laurence R. Bentley BSTRT orosty, permeablty and other parameters must be specfed at every node wthn a petroleum reservor smulator. The parameters are developed from sparse and nosy data so that they are not known exactly. fter prelmnary assgnment of porosty and permeablty data, smulators are run n the forward mode, but they can seldom match the observed producton hstory adequately. onsequently, a hstory matchng step s requred n whch porosty, permeablty and other parameters are adjusted untl the model predctons match the producton hstory to a suffcent degree. In automatc hstory matchng, an objectve functon s formed by squarng and weghtng the dfference between observatons and the computed predctons. In addton, terms are often added to the objectve functon to penalze departures from orgnal parameter estmates. Ths objectve functon s then mnmzed wth respect to the parameters usng soluton technques such as the Marquardt method. Tme-lapse sesmc surveys produce mages at dfferent tmes n a reservor s hstory. The sesmc response of a reservor may change due to changes n pressure, flud saturaton and temperature. Gven approprate rock physcs models, the output of the reservor smulator can be used to predct the change n sesmc response. The dfference between the predcted change n sesmc response and the observed change n sesmc response forms another set of resduals. These resduals are squared, weghted and added to the objectve functon. INTRODUTION Reservor smulators are used to approxmately solve the mathematcal equatons that descrbe the physcs of flow of fluds wthn petroleum reservors. The smulators calculate changes n pressure and saturaton wthn reservors due to producton and the njecton of fluds durng secondary recovery operatons. Reservor smulators are used to forecast producton, assess rsk and test conceptual models. The mathematcal models used n reservor smulators requre the specfcaton of reservor parameters at all locatons wthn the reservor. In partcular, the ntrnsc permeablty and the porosty need to be specfed. The values of these attrbutes are extremely heterogeneous n geologc formatons, and they exert strong control on the flow and storage characterstcs of the reservor. In practce, these parameters are known approxmately, because hard data from wells are sparse, correlaton to densely spaced sesmc data s typcally weak and dfferences exst between measurement and modelng scales. In general, forward smulatons usng these values wll not adequately match the observed petroleum reservor producton hstory on the frst attempt, because of the uncertanty n the ntal estmates of the parameters. onsequently, the parameters REWES Research Report Volume 10 (1998) 33-1

2 ontents Bentley are teratvely adjusted untl an adequate match of the producton hstory s obtaned. Ths process s known as hstory matchng utomatc hstory matchng (M) s a process n whch hstory matchng s performed usng optmzaton technques. In ths approach, an objectve functon s formed whch s a functon of the msmatch between the calculated values of pressure and producton hstory and the observed pressure and producton hstory. The objectve functon s then mnmzed wth respect to the parameters to obtan an updated set of parameters for use n the smulator. DEVELOMENT Notaton vector of true producton hstory (pressure, water cut, gas ol rato, etc.) vector of measured producton hstory vector of computed producton hstory vector of computed producton hstory at the optmzaton pont N number of producton hstory vector entres e vector of errors assocated wth σ covarance of e covarance of e scaled by 1/σ σ common varance of e W producton hstory weght r producton hstory resdual vector X roducton hstory senstvty matrx vector of true sesmc attrbutes vector of measured sesmc attrbutes vector of computed sesmc attrbutes vector of computed sesmc attrbutes at the optmzaton pont N number of sesmc attrbute vector entres e vector of errors assocated wth 33- REWES Research Report Volume 10 (1998)

3 ontents On ncorporatng tme-lapse sesmc survey data σ covarance of e covarance of e scaled by 1/σ σ common varance of e W sesmc attrbute weght r sesmc attrbute resdual vector X sesmc attrbute senstvty matrx vector of true parameters values vector of measured parameter values (pror nformaton) vector of current parameter values derved from the optmzaton routne vector of parameter values at the optmzaton pont N number of pror nformaton vector entres e vector of errors assocated wth σ covarance of e covarance of e scaled by 1/σ σ common varance of e W pror nformaton weght r pror nformaton resdual vector X pror nformaton senstvty vector n number of decson varables, n N p reservor pressure S O ol saturaton S W water saturaton S G gas saturaton K objectve functon K objectve functon at soluton REWES Research Report Volume 10 (1998) 33-3

4 ontents Bentley X total senstvty (acoban) matrx X total senstvty matrx at soluton r augmented resdual vector r augmented resdual vector at soluton G gradent vector e approxmate essan matrx λ M Marquardt parameter λ R step sze relaxaton parameter s common varance Formulaton onsder three objectve functon terms assocated wth the producton hstory, sesmc attrbutes and pror knowledge of the parameter values. Typcally the decson parameters,, wll consst of the log-permeablty and the porosty. The producton values,, are calculated usng the smulator and the model parameters. set of rock physcs models, such as the Gassman equaton, are used wth the pressure and saturaton values computed by the flow smulator to calculate the sesmc attrbutes,. roducton story ssume that producton hstory measurements are related to the real producton hstory by, + Further, the form of e s assumed to be normal wth zero mean and covarance σ. The matrx has the form of the covarance of e, but t s scaled by the nverse of the common varance σ. The most common models for s that the producton measurement errors are uncorrelated n space but may be correlated by a one lag autocorrelaton functon n tme. Ths means that s a dagonal or block dagonal matrx. The dagonal terms of would vary f, for example, one set of pressure data was consdered less relable than another set of pressure data. e (1) 33-4 REWES Research Report Volume 10 (1998)

5 ontents The producton hstory objectve functon term s, On ncorporatng tme-lapse sesmc survey data t 1 ( ) ( ). 1 1 σ σ () 1 The weghtng terms σ and -1 : 1. Scale the equaton for varatons n unts (e.g. ps versus Kpa),. Weght more relable data more than less relable data, and 3. Flter the estmates based on the correlaton between errors. Sesmc ttrbutes In a manner analogous to the producton hstory match, the sesmc attrbute match leads to, t 1 ( ) ( ). 1 1 σ σ (3) In ths case, t s possble the e may be correlated and the optmal would be a non-dagonal matrx. ror Estmates of arameters Estmates of porosty have been developed usng well log nformaton or well log nformaton n conjuncton wth sesmc attrbutes. ermeablty estmates may be derved from upscalng core permeablty, DST analyss, and correlaton of porosty wth permeablty. The nformaton contaned n these pror estmates can be represented by the objectve functon term, t 1 ( ) ( ). 1 1 σ σ (4) In ths case, the form of the covarance of e Y may be determned from the estmaton errors generated n the nterpolaton step (e.g. krgng or cokrgng). Ths term s sometmes referred to as a regularzaton term and mproves the stablty of nverse procedures. In general, e wll be correlated, and usng the full correlaton structure whle developng the weghtng matrces has been shown to mprove estmates (Bentley, 1997). Optmzaton ssumng that the statstcal assumptons (normalty, etc.) are correct and that near the soluton the objectve functon s nearly lnear n the parameters, then the followng objectve functon s optmal n a least squares sense, REWES Research Report Volume 10 (1998) 33-5

6 ontents Bentley MIN 1... σ σ σ W R T 1 The correct weghtng s seen to be the nverse of the covarance of errors n the measurements and pror estmates. Usng ths form requres a pror knowledge of the covarance of the errors, whch are not known. In addton, we have neglected model error n the sense that the flow smulator may not be able to match the true system response due to nadequaces n the model equatons or dscretzaton. The assumpton of normally dstrbuted errors leads to problems f there are extremely large resduals. To date, mnmzaton of the equvalent of the full objectve functon 1 has not been reported. Researchers have optmzed the equvalent of σ σ (e.g. Gavalas et al. 1976, hu et al. 1995, havent et al. 1975), or the equvalent of (uang, et al. 1997, uang et al. 1998). In addton, Landa σ σ and orne (1997) use tme lapse sesmc data to estmate the change n saturaton wthn the reservor and then solved a form of. n approach that has been used n water resources s to assume that we know the form of the covarance matrx, but not the magntude of the common varances (arrera and Neuman, 1986). In ths way, nformaton on the relatonshp between the errors wthn each resdual type s ncluded n the objectve functon, but the weghts assocated wth the common varances must be selected by some other crtera. These weghts determne the relatve mportance of each type of nformaton. The revsed producton hstory objectve functon term becomes, W W + 1 t 1 ( ) ( ) Smlar expressons are used for wth weghtng W and wth weghtng W. In the lterature the weghts W K (where K, or ) have been chosen by a varety of methods. The new objectve functon becomes, W + W + W 4 One method that has been used wth success s to vary the weghts based on balancng the weghted varances as approxmated by the current estmate of the parameters. On the assumpton that at the soluton, the values of, and are close to ther true values, the weghted varances of,, and can be approxmated by, + 1 (5) (6) (7) 33-6 REWES Research Report Volume 10 (1998)

7 ontents Var On ncorporatng tme-lapse sesmc survey data W t 1 ( ) ( ' ) ( ' ) Var Var N W t 1 ( ) ( ' ) ( ' ) N W t 1 ( ) ( ' ) ( ' ) N The ndex ndcates values derved from the soluton at teraton, and N, N and N are the number of pressure measurements, sesmc attrbute values and pror estmates of parameters, respectvely. In order to balance the contrbutons of the dfferent data types, new weghts are chosen to balance the weghted varances, W 1 K K Var W ( K ) 1 The weghts are adjusted at the end of each run untl the varances are approxmately equal. Two or three teratons are generally requred to reach a balanced set of weghted varances (Wess and Smth, 1998). omputatonal Issues Gradent and Senstvty Matrx Several approaches have been used to solve the mnmzaton problem (e.g. equaton (5)). In the followng, we focus on the Marquardt method (Marquardt, 1963). Each Marquardt teraton, the gradent and the senstvty coeffcents must be computed. Defne the resdual vectors, r r r The resdual vectors at the soluton are denoted r K, where K, or. The global resdual vector s t r t t t [ r r r ] The producton hstory resdual vector contans N entres. These wll be arranged as tme seres data from several to tens of producton and njecton wells. It s possble for sesmc attrbute data to exst at each grd block wthn the smulator, and consequently, the dmenson of the vector can theoretcally be n the tens of (8) (9) (10) (11) (1) (13) (14) (15) REWES Research Report Volume 10 (1998) 33-7

8 ontents Bentley thousands. Wth three permeablty values (κ X, κ Y, κ Z ) and a porosty assocated wth each grd block, the number of parameters can be n the tens of thousands. senstvty coeffcent s the partal dervatve of an ndvdual resdual wth respect to a decson varable. Each type of resdual has a senstvty matrx assocated wth t. n element of the producton hstory senstvty matrx s, X j where each row 1 to N s assocated wth a producton hstory resdual and each column j1 to n s assocated wth a decson parameter. Smlarly, the sesmc attrbute resdual senstvty coeffcents are, X j j where 1 to N and j1 to n. ssume that varatons n sesmc attrbute can be calculated from the pressure and saturaton, j (16) (17) F( p, SO, SW, SG ) Then, the sesmc attrbute senstvtes can be calculated usng the chan rule, (18) F p F + p S O SO F + S W SW F + S G SG onsequently, the partal dervatve of the pressure and saturatons wth respect to each parameter s requred at each locaton of a sesmc attrbute value. These same dervatves are requred for the evaluaton of equaton (16), but only at the locatons of the wells used n the producton hstory match. In some cases, temperature would need to be modeled as a state varable, because t can also have a sgnfcant effect on the sesmc response. The senstvty coeffcents for the pror nformaton are, X j wth rows 1 to N and columns j1 to n. The pror nformaton matrx conssts of entres of ones and zeroes, and f every parameter has pror nformaton then the matrx s the dentty matrx. j (19) (0) 33-8 REWES Research Report Volume 10 (1998)

9 ontents The total senstvty matrx s constructed, On ncorporatng tme-lapse sesmc survey data t X t t t [ X X X ] and t has NN +N +N rows and n columns. It s possble for the number of parameters to be n the tens of thousands. The huge dmensonalty of the parameter space and the lmted amount of observatonal data lead to a problem that s nherently underdetermned. onsequently, unqueness and dentfablty are ssues n the parameter optmzaton problem. In practce, the dmenson of the parameter space s reduced by lumpng many grd blocks together n zones of equal value or to use nterpolaton methods such as the plot pont method to populate the reservor wth a smaller number decson varables located ate the nterpolaton ponts (plot ponts). We are currently nvestgatng the use of cluster algorthms for dynamc zonng. The calculaton of the senstvty matrx s a major computatonal burden. Three approaches have been used. In the perturbaton method, each parameter s perturbed and a forward smulaton s run to compute the new value of the resduals. The senstvty matrx s then constructed by the fnte dfference method. Ths method s only computatonally feasble wth a small number of parameters. In the adjont equaton method, the equvalent of a separate smulaton run s needed for each of the observaton locatons. Ths method was useful when the number of observaton locatons was lmted to a small set of producton and njecton wells. owever, the ncluson of spatally dense sesmc attrbute data requres the senstvty calculatons to be carred out at thousands of locatons, and the adjont equaton method s not as attractve for the cases n whch tme-lapse sesmc data s ncluded n the objectve functon. In the gradent smulaton method (nteron et al. 1989), the equvalent of an extra flow smulaton plus for each decson varable one back substtuton s requred for each Marquardt teraton. t present, ths appears to be the best approach for calculatng senstvty coeffcents for the cases wth tme-lapse sesmc terms n the objectve functon. The global weghtng matrx for objectve functon 4 s, (1) 1 W W W The other objectve functons have analogous weghtng matrces. The gradent of the objectve functon s, () G X k t 1 where k ndcates the form of the objectve functon (.e. k1,, 3, or 4). The vector G has a length of n. r (3) REWES Research Report Volume 10 (1998) 33-9

10 ontents Bentley If the resduals are small and/or the equaton s quas-lnear, then the essan matrx can be approxmated as, e The matrx e has dmensons n by n. k t 1 ( X X ) (4) The Marquardt algorthm update s, t 1 k ( X X ) + I ) G k + 1 k k 1 + λr λm where k s the teraton ndex, λ M s the Marquardt parameter and λ R s a relaxaton parameter to control the step sze. Transformaton and Scalng Improvng the condtonng of the objectve functon wll mprove the soluton and reduce the computatonal effort requred to reach a soluton. arameters and resduals are often scaled or transformed n order to provde a better condtoned objectve functon. In addton, transformed varables often have better statstcal propertes than the prmary varables. For example, permeablty s generally transformed to logpermeablty for geologc and statstcal reasons. pproprate weghts wll depend on the form of the scaled or transformed varables, so transformaton choces are related to weghtng choces. Often the decson varables are normalzed to one n order to mprove the condtonng of the objectve functon. If the current value of the parameters s, then the scalng takes the form, (5) S ej S ej G G S j (6) (7) (8) Error Estmates t the soluton, the common varance of the weghted errors can be estmated by lnear error analyss, s ( N ' 4 + N + N ' 4 ) n N where N D s the dfference between the number of resduals and the number of decson varables. D (9) REWES Research Report Volume 10 (1998)

11 ontents n estmate for the estmaton varance of the parameters s, Var On ncorporatng tme-lapse sesmc survey data t 1 ( ' ) s ( X ' X ') 1 where the prme ndcates that the values are evaluated at the optmal soluton. lnear approxmaton of the confdence regons s gven by, (30) t t 1 ( ' ) X ' X '( ') n s F ( n, N ) where F s an F dstrbuton wth n and N D degrees of freedom. α D (31) ONLUSION The goal of automatc hstory matchng (M) s to mprove the parameter selecton for petroleum reservor smulators n order to mprove ther reservor forecastng capablty. n approach for ncludng tme-lapse sesmc data nto M has been presented and some practcal ssues have been dscussed. M s stll not routnely used, even after more than twenty years of development. The dffcultes are related to the extreme nonlnearty of the objectve functon, the hgh dmensonalty of the decson varable set, problems assocated wth unqueness and dentfablty of parameters, and dffcultes assocated wth computng the senstvty matrces. M based solely on producton data observed at a lmted number of well locatons suffers because of the sparsty of the spatal sample. Sesmc data has the nherent advantage of beng a spatally dense data set, and prelmnary ndcatons are that ncludng tme-lapse sesmc data wll lead to mproved M. lthough the routne applcaton of M remans an elusve goal, studyng tmelapse sesmc survey results n the context of M s valuable. The mathematcal formulaton of M gves us nsght nto the hstory matchng process and helps analysts dentfy the underlyng structure of the hstory match problem. Study of the senstvty of sesmc attrbutes to changes n reservor parameters, objectve functon gradents and estmaton varance wll lead to nsghts nto the effectveness of ntegratng tme-lapse sesmc data nto reservor smulaton and reservor forecastng. In turn, these nsghts wll gude tme-lapse sesmc acquston and processng choces. REFERENES nteron, F., Eymard, R., Karcher, B., 1989, Use of parameter gradents for reservor hstory matchng, SE Bentley, L.R., 1997, Influence of the regularzaton weghtng matrx on parameter estmates, dv. Wat. Resour., 0(4), arrera,. and Neuman, S.., 1986, Estmaton of aqufer parameters under transent and steady state condtons: 1. Maxmum lkelhood estmaton ncorporatng pror nformaton, Wat. Resour. Res., (), havent, G., Dupuy, M. and Lemonner,., 1975, story matchng by use of optmal theory, Soc. et. Eng.., 15(1), REWES Research Report Volume 10 (1998) 33-11

12 ontents Bentley hu, L., Reynolds,.. and Olver,.S., 1995, Reservor descrpton from statc and well-test data usng effcent gradent methods, SE 9999, Gavalas, G.R., Shah,.. and Senfeld,.., 1976, Reservor hstory matchng by Bayesan estmaton, Soc. et. Eng., 16(6), uang, X., Mester, L. and Workman, R., 1998, Improvement and senstvty of reservor characterzaton derved from tme-lapse sesmc data, SE uang, X., Mester, L. and Workman, R., 1997, Reservor characterzaton by ntegraton of tmelapse sesmc producton data, SE Landa,.L. and orne, R.N., 1997, procedure to ntegrate well test data, reservor performance hstory and 4-D sesmc nformaton nto a reservor descrpton, SE Marquardt, D.T., 1963,. Soc. Ind. ppl. Math., 11, Wess, R. and Smth, L., 1998, arameter space methods n jont parameter estmaton for groundwater flow models, Wat. Resour. Res., 34(4), REWES Research Report Volume 10 (1998)