assessing the uncertainty in reservoir description and
TRANSCRIPT
SPE 95750
Assessing the Uncertainty in Reservoir Description and Performance Predictions With theEnsemble Kalman Filter
Copyright 2005, Society of Petroleum Engineers
This paper was prepared for presentation at the 2005 SPE Annual Technical Con-ference and Exhibition held in Dallas, Texas, U.S.A., 9 - 12 October 2005.
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AbstractRecently, the ensemble Kalman filter (EnKF) has gainedpopularity in atmospheric science for the assimilationof data and the assessment of uncertainty in forecastsfor complex, large-scale problems. A handful of papershave discussed reservoir characterization applications ofthe EnKF, which can easily and quickly be coupled withany reservoir simulator. Neither adjoint code nor spe-cific knowledge of simulator numerics is required for im-plementation of the EnKF. Moreover, data are assimilated(matched) as they become available; a suite of plausiblereservoir models (the set of ensembles) is continuously up-dated to honor data without rematching data assimilatedpreviously. Because of these features, the method is farmore efficient for history matching dynamic data than au-tomatic history matching algorithms based on optimiza-tion algorithms. Moreover, the suite of ensembles providesa way to evaluate the uncertainty in reservoir descriptionand performance predictions.
Here we establish a firm theoretical relation be-tween randomized maximum likelihood and the ensembleKalman filter. We also consider examples where the per-formance of the EnKF does not provide a reliable char-acterization of uncertainty in the model or performancepredictions. Although we have previously generated reser-voir characterization examples where the method workedwell, here we also provide examples where the performanceof EnKF does not provide a reliable characterization of un-certainty.
IntroductionOur main interest is in characterizing the uncertainty inreservoir description and reservoir performance predictionin order to optimize reservoir management. To do so, wewish to generate a suite of plausible reservoir models (real-izations) that are consistent with all information and data.If the set of models is obtained by correctly sampling of thepdf, then the set of models give a characterization of theuncertainty in the reservoir model. By predicting futurereservoir performance with each of the realizations, andcalculating statistics on the set of outcomes, one can eval-uate the uncertainty in reservoir performance predictions.Uncertainty can only be formally characterized using themathematics of probability and statistics.
For data integration problems of interest in reservoirmodelling and characterization, Bayesian statistics pro-vides a convenient framework for characterizing and eval-uating uncertainty. The method introduced into reser-voir characterization by Oliver et al.1 and also consideredbriefly by Kitanidis2, which is now most commonly referredto as randomized maximum likelihood, has frequently beenused to generate an approximate sampling of pdf for areservoir model conditional to production and/or seismicdata3,4,5. However, this method often takes on computa-tional equivalent of about 100 reservoir simulation runs togenerate a single plausible reservoir model (realization orensemble) and its implementation requires efficient adjointcode which is not publicly available. In contrast to this,the EnKF method developed by Evensen6,7,8 and intro-duced into the petroleum engineering literature by Naev-dal and coworkers9,10 requires only one reservoir simula-tion run per ensemble. Here, we show that EnKF andRML become equivalent as the number of ensembles goesto infinity provided that one begins with a prior multi-variate Gaussian distribution for m, data are uncorrelatedin time and the relation between predicted data and thereservoir model is linear. No such equivalence is avail-able in the nonlinear case, although EnKF can be relatedto simple co-kriging and can also be related to using theGauss-Newton method when generating realizations in theRML method11. Because a Gaussian prior model is effec-tively assumed at each assimilation step of EnKF, it isnatural to question whether the method yields a reliablecharacterization of uncertainty in the case where the condi-tional pdf for the reservoir model is multi-modal, whereas,
M. Zafari, SPE, and A.C. Reynolds, SPE, U. of Tulsa
2 SPE 95750
one expects that RML at least generates realizations nearseveral modes even if it does not sample correctly1. Nev-ertheless, for the well known PUNQ example, Gao et al.12have shown that RML and EnKF both give reasonable es-timates of the uncertainty in performance predictions.
Our focus here is on examining the shortcomings ofEnKF by presenting examples where its performance interms of characterizing uncertainty is not completely sat-isfactory. We also investigate whether a reliable metric fordetermining when EnKF fails can be defined but our re-sults in this regard are not completely satisfactory. Finally,we investigate whether the mean of the ensembles providesa realiable estimate of the truth or whether a better esti-mate can be obtained.
Ensemble Kalman FilterWe consider both the model m and the predicted data,g(m), as random vectors and define a random vector, y, as
y =[
mg(m)
]. (1)
All vectors are column vectors with the dimension of mgiven by Nm and the dimension of g(m) is denoted byNd. Thus, the dimension of y, denoted by Ny is given byNy = Nm +Nd. In the implementation of EnKF, g(m) de-notes data predicted at a specific time tl corresponding tothe vector of observed data we wish to assimilate. If mea-sured data at all times were assimilated (matched) simul-taneously, then the vector dobs would include all measureddata and g(m) would represent the corresponding vector ofpredicted data. In reservoir characterization problems ofinterest to us, m represents reservoir variables such as grid-block porosities and permeabilities or log-permeabilitiesand g(m) is obtained by running a simulator with themodel m as input.
A critical step in the EnKF is that the assumptionthat the pdf for y can be approximated by the multivari-ate Gaussian distribution given by
f(y) = a exp(− 1
2(y − y
)TC−1
Y
(y − y
)), (2)
where a is the normalizing constant, y is the mean and CY
is the covariance matrix for y given by
CY =[
CM CM,G
CG,M CG
]. (3)
Throughout, CM is the autocovariance matrix for m, CG
is the autocovariance matrix for g(m) and CM,G and CG,M
are the cross covariance matrices for m and g(m), i.e.,
CM = E[(
m− m)(
m− m)T ]
, (4)
CG = E[(
g(m)− g(m))(
g(m)− g(m))T ]
, (5)
CM,G = CG,M = E[(
m− m)(
g(m)− g(m))T ]
, (6)
If measurement error plus modelling error is a Gaussianrandom vector with mean given by the Nd-dimensionalzero vector and covariance given by CD, a standard ap-plication of Bayes’ theorem gives that the posterior pdf fory conditional to dobs is given by
f(y|dobs) = a exp(− 1
2(g(m)− dobs
)TC−1
D
(g(m)− dobs
))exp
(− 1
2(y − y
)TC−1
Y
(y − y
)).
(7)
where a is now the normalizing constant for the condi-tional pdf f(y|dobs). Recall that g(m) and dobs are Nd-dimensional column vectors. Let INd
denote the Nd ×Nd
identity matrix, O denote the Nd × Nm null matrix anddefine the Nd ×Ny matrix H by
H =[O INd
]. (8)
It follows from Eq. 1 that data predicted for a given y islinearly related to y since predicted data is given by
g(m) = Hy, (9)
and Eq. 7 can be written as
f(y|dobs) = a exp(− 1
2(Hy − dobs
)TC−1
D
(Hy − dobs
))exp
(− 1
2(y − y
)TC−1
Y
(y − y
)).
(10)
It is well known13 that the randomized maximum likeli-hood method, RML, can be applied to obtain a correctsampling of the posterior pdf, f(y|dobs) given by Eq. 10.
To generate one realization using RML, we generate aset of Ne unconditional realizations of y from the Gaussianpdf of Eq. 2. Such an unconditional realization, yuc,j , canbe obtained by generating an Ny-dimensional vector Zj ofindependent standard random normal deviates and calcu-lating
yuc,j = y + LyZj , (11)
whereCY = LyLT
y , (12)
is the Cholesky decomposition of CY . For each uncondi-tional realization muc,j , 1 ≤ j ≤ Ne, we generate a set ofrealization of the data, duc,j by adding noise to the datai.e.,
duc,j = dobs + LDZD,j , (13)
where ZD,j is a vector of standard random normal deviatesand LDLT
D is the Cholesky decomposition of the measure-ment error covariance matrix, CD. For each j, we minimize
Oj(y) =12(Hy − duc,j
)TC−1
D
(Hy − duc,j
)+
12(y − yuc,j
)TC−1
Y
(y − yuc,j
).
(14)
Then it is well known that the set of minima, yuj , 1 ≤ j ≤
Ne represents a set of samples from the conditional pdf
SPE 95750 3
f(y|dobs). The minimum of Oj(y) can be found by settingits gradient with respect to y equal to the Ny-dimensionalzero vector and solving for y. Denoting this solution byyu
j , we find that
yuj = yuc,j + CY HT
(CD + HCY HT
)−1(duc,j −Hyuc,j
).
(15)The preceding equation is effectively the basic equation forgenerating a suite of realizations using EnKF except yuc,j
is generated as data are assimilated sequentially in timeand the data part of each yuc,j is generated by using pre-diction from the forward model and CY is generated by anapproximate method.
Gu and Oliver14 were able to relate the MAP estimateto the ensemble Kalman filter. Here, we extend their re-sult to show that if data are linearly related to the model,the prior model is multivariate Gaussian, and data are un-correlated in time, then the generating ensembles usingEnKF is equivalent to sampling with randomized maxi-mum likelihood1 as the number of ensembles becomes in-finite.
Next we establish conditions under which the preced-ing procedure is equivalent to using RML to sample the pdff(m|dobs), the conditional pdf for m given dobs. We con-sider the case where (i) the prior model for m is Gaussianwith mean mprior and covariance matrix CM and (ii) pre-dicted data is linearly related to the model by
g(m) = Gm. (16)
From Eq. 16, the expectation of Gm is given by Gmprior
so
y =[
mprior
Gmprior
]. (17)
and Eq. 3 can be rewritten as
CY =[
CM CMGT
GCM GCMGT
]. (18)
Letting CM = LLT be the Cholesky decomposition of CM ,and defining the block lower triangular matrix Lc by
Lc =[
L OGL O
], (19)
where the O’s denote null matrices, it follows easily fromEq. 18 that LcL
Tc = CY , thus, Lc = Ly. Using this result
and Eq. 17 in Eq. 11 gives
yuc,j =[
muc,j
g(m)uc,j
]=
[mprior + LZM,j
G(mprior + LZM,j
)] , (20)
where ZM,j denotes the Nm dimensional vector with en-tries identical to the corresponding components of the firstNm dimensional entries of Zj . The vector consisting of thefirst Nm components of yuc,j pertains to the model m andis given by
muc,j = mprior + LZM,j . (21)
which represents a correct sampling of the prior Gaussiandistribution for m. From Eqs. 8, 18 and 15, it is straight-forward to show that the model part of yu
j is given by
muj = muc,j+CMGT
(CD+GCMGT
)−1(duc,j−g(muc,j)
).
(22)Eqs. 21 and 22 imply that the mu
j ’s extracted from theyu
j ’s are equivalent to applying RML to sample f(m|dobs)when predicted data is linearly related to m and the priorpdf for m is N(mprior, CM ). When the prior model isN(mprior, CM ), f(m|dobs) is given by
f(m|dobs) = a exp(− 1
2(g(m)− dobs
)TC−1
D
(g(m)− dobs
))exp
(− 1
2(m−mprior
)TC−1
M
(m−mprior
)),
(23)
where in the linear case, g(m) = Gm.In the case where dobs contains all dynamic data we
wish to assimilate, then for the purposes of uncertaintyevaluation in the model and future predictions using theforward model, our objective is to sample from f(m|dobs).To summarize our results, we have shown that the set ofyu
j ’s (Eq. 15) generated by RML always represents a cor-rect sampling of the pdf of Eq. 10 or equivalently Eq. 7.When the conditional pdf f(m|dobs) is given by Eq. 23 andg(m) = Gm, then the model part of the yu
j ’s represents acorrect sampling of f(m|dobs). If the observed data, dobs,can be written as
dobs =
dobs(t1)dobs(t2)
...dobs(tn)
, (24)
where dobs(t`) represents data measured at time t` andt1 < t2, · · · , < tn, and individual data are uncorrelatedin time, then generating the conditional mean of m (theMAP estimate) by minimizing is equivalent to generating asequence of MAP estimates where at the `th step in the se-quence, we minimize an objective function which includesonly the data mismatch between dobs,` and the correspond-ing predicted data15. After each minimization, the covari-ance matrix must updated to the a posteriori covariancematrix, CM,`. The MAP estimate based on assimilatingdobs,` and the covariance matrix CM,` effectively providethe prior Gaussian model for the assimilation of the nextset of data, dobs,`+1. Using these ideas, it can be shownthat RML can be used to assimilate data sequentially intime11 and that the unconditional realizations obtainedat the final time step tn represents a correct sampling off(m|dobs) provided again that (i) the prior pdf for m usedto assimilate data at the first time t1 is N(mprior, CM ),(ii) g(m) = Gm and (iii) data measurement errors are un-correlated in time and satisfy Gaussian distributions withzero means. Under these conditions, application of Eq. 15
4 SPE 95750
sequentially in time results in a set of yuj ’s after assimila-
tion at tn which reprsent a correct sampling of the pdf inEq. 10 and the model parts of the yu
j ’s represent a correctsampling of f(m|dobs) where, dobs is given by Eq. 24.
From this point on we assume that Eq. 15 is appliedsequentially in time that is duc,j represents data at sometime t` where our vector of observed data is dobs(t`). For1 ≤ j ≤ Ne, we let mp
j denote the model part of the en-semble vector obtained by assimilating data at t`−1, andletting g(m) now denote predicted data corresponding todobs = dobs(t`), it makes sense to replace yuc,j in Eq. 15 by
ypj =
[mp
j
g(mpj )
], (25)
so that Eq. 15 becomes
yuj = yp
j +CY HT(CD +HCY HT
)−1(duc,j −Hyp
j
), (26)
for 1 ≤ j ≤ Ne and replace y in the prior model by theaverage of the yp
j ’s. This is the natural analogue of the pro-cedure we established in the linear case. If we also estimatethe covariance CY by
CY =1
Ne − 1
Ne∑j=1
(yp
j − y)(ypj − y)T , (27)
and include time dependent parameters in y, then Eq. 15is mathematically equivalent to the basic EnKF algoirithmfor assimilating data. The approximation of Eq. 26 may berank deficient and so even in the linear case, we can onlyshow that EnKF is equivalent to RML and hence gives acorrect sampling as Ne →∞. The EnKF can also be mo-tivated from the Gauss-Newton method11 and from simpleco-kriging.
Updating of Time Dependent ParametersIn the presence of time dependent parameters, the statevector for jth ensemble will be defined by
yj =
mj
pj
dj
=[mT
j pTj dT
j
]T, (28)
where p is an Np-dimensional column vector and Np isthe number of time dependent parameters for each ensem-ble. If predicted data g(m) are generated from a black oilreservoir simulator, p includes pressure, saturations anddissolved GOR. Similar to predicted data, the uncondi-tional realizations of time dependent parameters are gen-erated by predicting these parameters for each ensemblemember. Here, we rewrite Eq. 28 in terms of matrix of allensembles,
Y =
MPD
, (29)
where M is an Nm×Ne matrix with the jth column equalto mj , P is an Np ×Ne matrix with the jth column equal
to pj and D is an Nd × Ne matrix with the jth columnequal to dj , predicted data from the jth ensemble. Here,we rewrite Eq. 26 in terms of the matrix of ensembles as
Y u = Y + CY HT (HCY HT + CD)−1(Duc −HY ). (30)
Note we have suppressed the superscript p, i.e., the j col-umn of the Y matrix which appears on the right side ofEq. 26 is equal to yp
j . Duc is an Nd ×Ne matrix with thejth column equal to duc,j . The approximation of Eq. 27 isequivalent to
CY ≈ 1Ne − 1
(Y − Y )(Y − Y )T , (31)
by substituting CY from the preceding equation into Eq. 30and simplifying, we obtain
Y u = Y(I +
∆DT
Ne − 1
[∆D∆DT
Ne − 1+ CD
]−1
(Duc −D))
= Y(I + δ(D)
) ,
(32)
where ∆D = D −D and
δ(D) =∆DT
Ne − 1
[∆D∆DT
Ne − 1+ CD
]−1
(Duc −D)). (33)
Note that M , P and D all get updated using the samecoefficient matrix,
(I + δ(D)
), where Eq. 32 indicates the
updating coefficient matrix is only a function of the ma-trix D. The idea is that by updating P as well as M , oneends up with values of time dependent variables and datathat are consistent with those that would be obtained if westarted with the updated models in M and ran the forwardmodel (reservoir simulator) from time zero to that assim-ilation time, i.e., old data are automatically honored be-cause they were incorporated as covariances were updated.This basic idea is well described in a Bayesian frameworkby Evensen8.
Because pressures and saturations computed from thereservoir simulator are replaced by their updated values(Eq. 32) at a time step where data are assimilated andthen used in continuing the reservoir simulation at thenext simulation time step, nonphysical values of pressureand saturation could be obtained. If this occurs, it seemsreasonable not to use them in future time steps, but toinstead redo the simulation time step using the updatedvalues of m. Wen and Chen16 used this type of proce-dure. While it seems very reasonable, we show in the nextsection that the rerun procedure is inconsistent with theunderlying theoretical motivation of EnKF for the simplelinear case.
Because of this, in the rare case that the updating stepgives an unreasonable value of time dependent variables,we simply truncate it to a reasonable value. For example,if an updated grid block saturation is negative, then wereplace it by zero.
SPE 95750 5
Linear Problem. Suppose we have the following rela-tionships between p, d and m,
pk+1 = Fkmk + Akpk + αk (34a)
dk+1 = Gkmk + Bkpk + βk (34b)
where Fk, Ak, Gk and Bk are matrices and αk and βk areknown vectors. Here, k denotes the time step, pk+1 anddk+1 are at a time tk+1 where we assimilate data. (Notethat if dk+1 is given as a function of mk and pk+1 we canstill obtain Eq. 34b which gives dk+1 as a function of mk
and pk by using a new set of coefficient matrices.) Werewrite Eqs. 34a and 34b, in the form of a matrix of en-sembles as
P k+1 = FkMk + AkP k + αk I , (35a)
Dk+1 = GkMk + BkP k + βk I , (35b)
where I is an Ne-dimensional row vector with each entryequal to 1.
Now suppose we have data at two time steps, k + 1stand k + 2nd time steps, that we wish to assimilate. Afterassimilation of data at tk+1, from Eqs. 32 and 35a, we seethat for the standard EnKF method where we update thetime dependent parameters, we have
P k+1,u = P k+1(I + δ(Dk+1)
)= FkMk
(I + δ(Dk+1)
)+ AkP k
(I + δ(Dk+1)
)+ αk I
, (36)
whereas, for the case that we rerun the time step, we have
P k+1,r = FkMk(I + δ(Dk+1)
)+ AkP k + αk I . (37)
Now we continue assimilation of data for one more timestep. After assimilation of second set of data at time tk+2,the normal EnKF updating gives
P k+2,u = Fk+1Mk(I + δ(Dk+1)
)(I + δ(Dk+2)
)+ Ak+1
(FkMk
(I + δ(Dk+1)
)+
AkP k(I + δ(Dk+1)
)+ αk I
)(I + δ(Dk+2)
)+ αk+1I
(38)
For the case that we rerun the time step,
P k+2,r = Fk+1Mk(I + δ(Dk+1)
)(I + δ(Dk+2)
)+ Ak+1
(FkMk
(I + δ(Dk+1)
)+ AkP k + αk I
)+ αk+1I
(39)
Although the δ(Dk+2) in Eqs. 38 and 39 are not the same,by comparing the right hand sides of these equations, wecan see that in the second term on RHS of Eq. 39 the model
part has not been updated. If we use the final model para-meters obtained from the case where we reran the data as-similation time step and apply the forward model (Eqs. 35aand 35b) forward from “time zero”, the predicted P k+2 willnot be consistent with P k+2,r. Thus, it is clear that thererun option should not be used.
Toy Problem 1Here, we apply the EnKF to a simple non-linear problem.For this example, there is only one model parameter, i.e. mis a real random variable. Consider the following forwardmodel,
d = g(m, t) = 1− 92
(m− 2π
3
)2
+ (t− 1) sin(m), (40)
where m is a scalar model parameter and t is time. Tomake this model similar to a problem where our forwardmodel is a reservoir simulator, we define the following re-cursive equation for predicting data with the restart optionof the simulator,
g(m, t + ∆t) = g(m, t) + sin(m)∆t. (41)
Using the preceding equation is similar to using the restartoption in reservoir simulator. We generate true syntheticdata using the true model which is mtrue = 1.88358, atfive times t = 1, 2, ..., 5. At each time, there is a single da-tum. To generate synthetic observed data, we add normalrandom noise generated from N(0, 0.01) as measurementerror. We consider a normally distributed prior model,m ∼ N(2.4, 0.1). To assimilate the data at each time stepwe update the time dependent parameters along with themodel parameters during the assimilation of data. Notethat in this problem the data and time dependent parame-ter (p in Eq. 28) for each data assimilation time step arethe same. Fig. 1 shows the prior pdf and the posterior dis-tribution after conditioning to 1, 2, 3, 4 and 5 data. Theposteriori pdf’s are non-Gaussian and the pdf’s conditionalto data at t1, t2 and t3 have two distinct modes. Becauseof the Gaussian assumptions made in EnKF, this prob-lem should be one in which EnKF encounters difficulty. Inthis case, we start the process with 5001 ensembles, 5000generated as unconditional realizations from the prior plusone ensemble equal to the prior mean. Figs. 2 through 4show the histogram of the model parameter and the pos-terior pdf after assimilation of the data at t = 1, 3 and 5.Here, we add the prior mean to the set of ensembles andduring data assimilation, update this ensemble by assimi-lating dobs (not a duc) to obtain an updated model whichwe refer to as the central model. For the linear case, itis easy to show that the central model is the same as theMAP estimate. In Fig. 2 the posterior pdf shows highprobability around the model m = 2.30521 (second root ofthe quadratic part of the model) because for t = 1 the sineterm in Eq. 40 does not have any effect on the predicteddata. By assimilating more data the sine term in the modelbecomes dominant and m = 2.30521 is no longer a valid
6 SPE 95750
model. Figs. 3 and 4 show how the posterior pdf moves tothe right model and finally has a single mode close to thetruth. As we can see, in this problem, the mean model isalways slightly closer to the true model for all data assimi-lation steps. After assimilation of the datum correspondingto t = 5, EnKF (see Fig. 4) gives a more reliable samplingof the correct pdf, but still gives a distribution which is toobroad and does not give as reliable characterization of thetrue pdf as is obtained with RML. (The RML results arenot shown.) Even as we continue to assimilate data be-yond t = 5, the distribution obtained with EnKF remainsbroader than the true distribution. Overall, the resultssuggest that the standard EnKF method does not give areliable estimate of the true pdf in a multi-modal case, i.e.,does not give a proper characterization of the uncertaintyin m and even in the single mode case may over estimatethe uncertainty in m. Although the results are not shownhere, using a rerun option to generate data results in aneven worse characterization of uncertainty11.
Toy Problem 2In previous problem, as we assimilate more data the poste-rior pdf moves toward a conditional pdf with a single mode.To investigate a problem, as more data are assimilated, theposterior pdf remains a multi-modal distribution, we applythe EnKF to the following simple non-linear problem. Theforward model is given by
d = g(m, t) = 1− 9t
2
(m− 2π
3
)2
. (42)
To make this data prediction similar to a problem whereour forward model is a reservoir simulator, we rewriteEq. 42 as,
g(m, t + ∆t) = g(m, t)− 9∆t
2
(m− 2π
3
)2
. (43)
We generate true synthetic data using the true modelwhich is mtrue = 1.88358, at five times t = 1, 2, ..., 5 (Noteeach datum which can be produced by the true model, canalso be generated by m = 2.30521). At each time, thereis a single datum. To generate synthetic observed data,we add normal random noise generated from N(0, 0.01) asmeasurement error. We consider a normally distributedprior model, m ∼ N(2.1, 0.2). To assimilate the data ateach time step we update the time dependent parametersalong with the model parameters during the assimilationof data. Fig. 5 shows the posterior pdf after assimilationof fifth data at t = 5. Fig. 6 shows the histogram of themodel parameter after assimilation of the data at t = 5.We can see the real posterior pdf is much narrower thanthe histogram generated by the ensembles. In this case,RML provides an almost perfect characterization of theset of conditional posterior pdf’s. This example confirmsagain that EnKF may provide a relatively poor character-ization of uncertainty in the model in a multi-modal case.For this example, both the mean and the “central model”
have zero probability and provide a poor estimate of thetruth.
Metric. Fig. 7 shows the difference between the averagemodel and the truth after assimilation of each data. Forthe first toy problem the difference decreases after eachdata assimilation but in the second toy problem the differ-ence between the average model, which we usually consideras our estimate from the EnKF method, is not becomingcloser to the truth as more data are assimilated. Fig. 8illustrate the average of the difference between each en-semble and true model, i.e., the average of |m − mtrue|.We can see for the second problem the average differencebetween the ensembles and the true model decreases asdata at t = 4 and t = 5 are assimilated. For toy problem1, EnKF clearly provides a better characterization of theconditional pdf for m as more data are assimilated, seeFigs. 2 through 4. This is also true in case 2 although theimprovement is slight and is not apparent from the resultswe have provided. The question is what metric should beused to quantify the reliability of EnKF in terms of provid-ing an estimate of the truth. The mean of the ensemblesis conventionally used as an estimate even though we haveshown that neither the mean nor the central model areguaranteed to provide a good estimate. This is consistentwith the fact that EnKF gives a slightly better character-ization of the relevant pdf as more data is assimilated andsuggests that the average of the difference between eachensemble and the truth is a better indicator of how EnKFis performing even if average model is used as the estimate.The results of Fig. 7 are based on using the mean modelas an estimator of the true model, but similar results areobtained if the central model is used as the estimator ofthe truth.
Although the results from the two toy problems showthat the central model is not a better estimator than themean model, this is not always true. Our experimentsfor the well known PUNQ-S3 problem has shown that thecentral model is a better estimator than the mean. Figs. 9through 11 show the log-permeability field for the thirdlayer of PUNQ-S3 problem. We can see the geologicalfeatures in central model are closer to the truth than themean model(For more information about applying EnKFon PUNQ-S3 example see Gao et al.12). Note the cen-tral model becomes more important in presence of timedependent parameters, i.e. after assimilation of produc-tion data, to predict the performance of the reservoir, thecentral model can be run by its updated time dependentparameters.
2D Synthetic ProblemThe horizontal 2D problem consists of a 20× 30 grid with∆x = ∆y = 300 feet and 600 active gridblocks. There are 6producing wells. The production consists of 90 days of pro-duction from well PRO-1 with all other wells shut in. Afterthis period well PRO-2 begins production while well PRO-1 continue to produce. This sequence is continued until we
SPE 95750 7
reach a total time of t = 540 days at which well PRO-6 hasbeen producing for exactly 90 days. The production rateat each well is specified to be qo = 100 STB/day up to 540days. After this 540 day period, we change the well PRO-1from a producer to an injector, INJ-1, and continue to pro-duce from the other wells until 5850 days. At day 5850, weclose well PRO-5 and continue production from the rest ofthe wells, until 7290 days. From t = 540 to 5850 days, thewater injection rate is 1000 STB/day and during this timeperiod, the wellbore constraint at the five producing wellsis qo = 200 STB/day. At t = 5850 days, the injection rateis increased to 2000 STB/day, well PRO-5 is shut in andthe other wells produce based on a target rate of qo = 300STB/day.
The true rock property fields were generated using se-quential Gaussian co-simulation. The key geostatisticalparameters used to generate the truth are listed in Ta-ble. 1. The production data generated consist of (i) flow-ing bottom hole pressure, producing gas-oil ratio (GOR)and water cut at producing wells; (ii) bottom hole pres-sure of the injector. We generated these true data usingthe ECLIPSE simulator. To generate the observed data,dobs, we added Gaussian noise to the true production data.The relative error added to the true data is 5% for bottomhole pressures, 5% for producing gas-oil ratio and 0.1% forwater cut.
Ensembles. A suite of unconditional realizations (ensem-bles) were generated using joint sequential multi-Gaussiancode developed by Gomez-Hernandez et al.17. For the en-semble Kalman filter, 90 unconditional realizations weregenerated as the initial set of ensembles based on the truegeostatistical parameters given in Table. 1.
ResultsFig. 12 shows the difference between the average modeland the true model for the porosity and log-permeabilityfields. Here we define the function Ψ by
Ψ(m) =1
Nm
Nm∑j=1
([
1Ne
Ne∑i=1
mij ]−mtrue
j
)2, (44)
where Nm is the number of a particular type of modelparameters and 1
Ne
∑Ne
i=1 mij equals the average of all en-
sembles for the jth grid block. Similar to the second toyproblem, the metric of Eq. 44 shows that we are not ob-taining more reliable estimate of the model as more dataare assimilated. But when we compare the 2D map ofthe porosity and log-permeability field with respect to thetruth, Figs. 14 through 19, we can see that there is an im-provement in estimating the structure of the true model.Therefore we define a new function Γ by
Γ(m) =1
Ne
Ne∑i=1
1Nm
Nm∑j=1
(mi
j −mtruej
)2, (45)
Fig. 13 shows the behavior of function Γ for both poros-ity and log-permeability after each data assimilation. We
can see that in general, this metric suggest that the en-sembles are getting closer to the truth as more data areassimilated. Note that Γ(m) shows a noticeable increaseat the two times (t = 540 and t = 5850 days) where thewell producing conditions were changed sharply.
Figs. 20 and 21 show the water saturation profile forthe true model and average of all ensembles after assimi-lation of data at 7290 days. We can see that there is goodagreement between the truth and the average of saturationfield. Calculations show that there is 5% error in the wa-ter content of the reservoir after 7290 days. The two blackregions on the average predicted saturation profile belongsto the places where the updated water saturation is lessthan irreducible water saturation.
Data Match and Performance Prediction. To inves-tigate the EnKF for predicting the future performance ofthe reservoir, first we predict the cumulative oil produc-tion for initial ensembles (it is possible that some of theensembles are not physically able to produce under thedesired conditions for the 10000 day period considered).Then we assimilate the observed data for 5400 days, andpredict until day 10000. In the next case we continue theassimilation of data until 7290 days, then predict until day10000. The reason that we use two different steps to assim-ilate the data is to see the effect of data assimilation in thelate production history. After 7290 days all the productionwells (well Pro-5 will remain shut in) will produce at con-stant bottom hole pressure constraint. The bottom holepressure constraint for wells PRO-2, 3, 4, and 6 are 2500,1000, 1500 and 300 psi respectively. The water injectionrate will remain 2000 STB/day as before.
Figs. 22 through 24 show the predicted cumulativeoil production after 7290 days for three cases (i)set of ini-tial ensembles (ii)after 5400 days of data assimilation and(iii) after 7290 days of data assimilation (before the day7290 all the wells are producing under the constant oilrate constraint therefore the cumulative oil production isa fixed number for all the ensembles except for some ofthe unconditional realizations that can not produce underthe constant oil rate constraint). Note that in second andthird cases after assimilation of production data the bandof predictions decreases significantly as compared to thepredictions from initial ensembles, which means that theassimilation of data has greatly reduced the uncertainty inthe production forecast. The results from the third caseshow that most of the ensembles over predict the cumu-lative oil production during the forecasting period. Thisoccurs when the water cut in the wells producing underthe constant bottom hole pressure constraint tends to belower than the true water cut.
Fig. 25 shows the match of bottom hole pressure forwell INJ-1(PRO-1 first 540 days) and prediction of bottomhole pressure until day 10000. Fig. 26 illustrate the samedata except we continue assimilation of data for 7290 daysthen predict until day 10000. We can see that by assimilat-ing the data between 5400 and 7290 days, the prediction is
8 SPE 95750
improved. In Fig. 25, after all the production wells switchto the constant bottom hole pressure constraint at t = 7290days the injection pressure for some of the ensembles startsto increase. This is because at t = 7290 days, for some ofthe ensembles the predicted bottom hole pressure in pro-duction wells are below the constant bottom hole pressureconstraint. This can be shown in Fig. 27 where at thetime 7290 days there are some ensembles where the pre-dicted bottom hole pressure for the well PRO-3 is belowthe constraint and the well will be shut in. Fig. 28 showsthe bottom hole pressure of the well PRO-3 after 7290 daydata assimilation. Here all of the ensembles are able toproduce under the bottom hole pressure constraint.
Figs. 29 and 30 show the match of production gas oilratio for the well PRO-3. Again we can see after assimila-tion of data between 5400 to 7290 days the ensembles re-sults in lower uncertainty in performance prediction. How-ever, the predicted data from the the central model doesnot change much.
In this problem during the early producing period, wa-ter break through does not occur in any of the wells. Attime 5400 days water break through has occurred only inwell PRO-5. Here, there is an interesting problem, at thetime of water break through in well PRO-5, only one of theensembles predict a significant water cut, therefore ∆D isalmost a null matrix (all the ensembles predict almost thesame value) and based on Eq. 32 there will be no correc-tion to the model parameters. Fig. 31 shows the water cutin well PRO-5, we can see there is almost no improvementin the water cut prediction before the predicted water cutsbecome significant in some of the ensembles. Figs. 32 and33 illustrate the water cut for well PRO-2 (the second pro-duction well in the high permeable region). Assimilationof more observations between 5400 days and 7290 days ingeneral improves the predicted water cut from the ensem-bles but during that period the predicted water cut fromthe central model seems to get worse.
Concluding RemarksThe Gaussian assumption of EnKF is very critical. Evenfor large number of ensembles, in some of the multi-modalproblems EnKF can not sample the posterior pdf correctly.For the linear case, we showed that EnKF samples cor-rectly from the posterior pdf as the number of ensemblesgoes to infinity and is equivalent to randomized maximumlikelihood method. However, for our two simple nonlinearproblem EnKF did not work as well as RML. While the en-semble mean is frequently used as estimation of the truth,the deviation of this mean from the truth does not give areliable metric for quantifying how well the posterior pdf ischaracterized by the distribution of the ensembles. For ourtwo toy problems central model is not a good estimate ofthe true model, but for the 2D synthetic problem the datapredicted from the central model is in a good agreementwith the truth and the central model provide a reasonableapproximation of future performance prediction. Although
during the late time of data assimilation the model para-meters start to diverge from the truth and many of theensembles over predict the ultimate oil recovery, the finalprediction performance is still reasonable. For the linearcase, we have shown that rerunning the time step to re-compute time dependent parameters is inappropriate.
Although, we have focused on problems where EnKFencounters some difficulties, it is likely that modificationswill be found to further improve its reliability. Moreover,the advantage of the method in terms of computationalefficiency and simplicity in implementation, suggest thatit will become widely used in practice even if it provides aless than perfect characterization of variability in reservoirvariables, fluid distributions and performance predictions.
Nomenclature
CD = covariance matrix of measurement errorsCG = covariance matrix of predicted dataCM = covariance matrix of model parametersCY = covariance matrix of state variablesD = matrix of datad = data vector
E[.] = expected valuef(.) = probability density function
G = sensitivity matrixg(.) = forward modelH = linear coefficient matrixM = matrix of model parametersm = vector of model parameters
Nd = number of dataNe = number of ensemblesNm = number of model parametersNp = number of time dependent parameters
O(.) = objective functionp = vector of time dependent parametersY = matrix of ensemblesy = vector of state variablesy = mean of vector of state variables
Zj = random vector
Greek
δ(.) = updating coefficient matrixε = measurement error
SPE 95750 9
Subscripts
obs = observedk, l = time index
j = ensemble member indexuc = unconditional realization
Superscripts
p = predictedT = transposeu = updated−1 = inverse
AcknowledgementsThe support of the member companies of TUPREP is
very gratefully acknowledged.
References1. Oliver, D.S., He, N. and Reynolds, A.C.: “Conditioning
Permeability Fields to Pressure Data,” European Con-ference for the Mathematics of Oil Recovery, V (1996).
2. Kitanidis, P.K.: “Quasi-linear geostatistical theory forinversing,” Water Resour. Res. (1995) 31, 2411.
3. Zhang, F. and Reynolds, A.C.: “Optimization Algo-rithms for Automatic History Matching of ProductionData,” Proceedings of 8th European Conference on theMathematics of Oil Recovery (2002) , 1.
4. Zhang, F., Skjervheim, J.A., Reynolds, A.C. andOliver, D.S.: “Automatic History Matching in aBayesian Framework: Example Applications (SPE-84461),” 2003 SPE Annual Technical Conference andExhibition (2003) .
5. Dong, Y. and Oliver, D.S.: “Quantitative use of 4D seis-mic data for reservoir characterization (SPE-84571),”Proceedings of 2003 SPE Annual Technical Conferenceand Exhibition (2003) .
6. Evensen, G.: “Sequential Data Assimilation with aNonlinear Quasi-Geostrophic Model using Monte CarloMethods to Forecast Error Statistics,” Journal of Geo-physical Research (1994) 99, 10143.
7. Evensen, G.: “The Ensemble Kalman Filter: Theoreti-cal Formulation and Practical Implementation,” OceanDynamics (2003) 53, 343.
8. Evensen, G.: “The Combined Parameter and State Es-timation Problem,” Computational Geosciences (2005), submitted.
9. Naevdal, G., Mannseth, T. and Vefring, E.H.: “Near-Well Reservoir Monitoring Through Ensemble KalmanFilter (SPE-75235),” Proceeding of SPE/DOE ImprovedOil Recovery Symposium (2002) .
10. Naevdal, G., Johnsen, L.M., Aanonsen, S.I. and Ve-fring, E.H.: “Reservoir Monitoring and ContinuousModel Updating Using Ensemble Kalman Filter (SPE-84372),” 2003 SPE Annual Technical Conference andExhibition (2003) .
11. Zafari, M. and Reynolds, A.C.: “EnKF Versus RML,Theoretical Notes and Computational Experiments,”TUPREP Research Report 22 (2005) , 195.
12. Gao, G., Zafari, M. and Reynolds, A.C.: “QuantifyingUncertainty for the PUNQ-S3 Problem in a BayesianSetting With RML and EnKF (SPE-93324),” SPEReservoir Simulation Symposium (2005) .
13. Oliver, D.S.: “On Conditional Simulation to InaccurateData,” Math. Geology (1996) 28, 811.
14. Gu, Y. and Oliver, D.S.: “The Ensemble Kalman Filterfor Continuous Updating of Reservoir Simulation Mod-els,” TUPREP Research Report 21 (2004) , 150.
15. Tarantola, A.: Inverse Problem Theory: Methods forData Fitting and Model Parameter Estimation, Else-vier, Amsterdam, The Netherlands (1987).
16. Wen, X.H. and Chen, W.: “Real-Time ReservoirModel Updating Using Ensemble Kalman Filter (SPE-92991),” SPE Reservoir Simulation Symposium (2005).
17. Gomez-Hernandez, J.J. and Journel, A.G.: “Joint Se-quential Simulation of MultiGaussian Fields,” Geosta-tistic Troia 92 , ed. A. Soares (1992) 133–144.
10 SPE 95750
φmean 0.2[ln k]mean 4.0
σφ 0.05σln k 2.0ρφ,ln k 0.8
α 40◦
r1 (ft) 8400r2 (ft) 1500
Table 1: Geostatistical parameters.
1 . 5 2 . 0 2 . 5 3 . 0 3 . 50123456789
1 01 11 2
f (m), f
(m|d)
m
P r i o r M o d e l 1 s t D a t a 2 n d D a t a 3 r d D a t a 4 t h D a t a 5 t h D a t a
Fig. 1: Prior pdf and posterior pdf after integrating 1, 2,3, 4 and 5 data.
1 . 5 0 1 . 7 5 2 . 0 0 2 . 2 5 2 . 5 00
1
2
3
4
5
6
f (m|d)
m
M o d e l S a m p l e s 1 s t P o s t e r i o r P D F C e n t r a l M o d e l M e a n M o d e l
Fig. 2: Posterior pdf and ensembles after assimilation offirst data, EnKF.
1 . 5 0 1 . 7 5 2 . 0 0 2 . 2 5 2 . 5 00123456789
1 0 M o d e l S a m p l e s 3 r d P o s t e r i o r P D F C e n t r a l M o d e l M e a n M o d e l
f (m|d)
m
Fig. 3: Posterior pdf and ensembles after assimilation ofthird data, EnKF.
1 . 5 0 1 . 7 5 2 . 0 0 2 . 2 5 2 . 5 00123456789
1 01 11 2
M o d e l S a m p l e s 5 t h P o s t e r i o r P D F C e n t r a l M o d e l M e a n M o d e l
f (m|d)
m
Fig. 4: Posterior pdf and ensembles after assimilation offifth data, EnKF.
SPE 95750 11
1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 60
5
1 0
1 5
2 0
2 5
3 0
f (m|d)
m
5 t h P o s t e r i o r P D F
Fig. 5: Posterior pdf after assimilation of fifth data.
1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 60 . 0
0 . 5
1 . 0
1 . 5
2 . 0
2 . 5
3 . 0
f (m|d)
m
S t a n d a r d E n K F , 5 t h D a t a
Fig. 6: Ensembles after assimilation of fifth data, EnKF.
0 1 2 3 4 50 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
| E(m
) - m tru
e |
T i m e
1 s t T o y P r o b l e m 2 n d T o y P r o b l e m
Fig. 7: Difference of the average of ensembles and truemodel after each data assimilation.
0 1 2 3 4 50 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5 1 s t T o y P r o b l e m 2 n d T o y P r o b l e m
E( | m
- mtru
e | )
T i m e
Fig. 8: Average of the differences between each ensembleand true model after each data assimilation.
12 SPE 95750
5 1 0 1 5 2 0
51 01 52 02 53 0
02 . 04 . 06 . 08 . 0
Fig. 9: PUNQ-S3 example, layer 3 truelog-permeability field.
5 1 0 1 5 2 0
51 01 52 02 53 0
02 . 04 . 06 . 08 . 0
Fig. 10: PUNQ-S3 example, layer 3average log-permeability field,EnKF.
5 1 0 1 5 2 0
51 01 52 02 53 0
02 . 04 . 06 . 08 . 0
Fig. 11: PUNQ-S3 example, layer 3central model log-permeabilityfield, EnKF.
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 02 . 5
3 . 0
3 . 5
4 . 0
4 . 5
5 . 0
5 . 5 Y � l n � Y ��������
Y����
���� , Y
�ln��
�������
Fig. 12: Difference of the average of ensembles and truemodel after each data assimilation.
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 03 . 54 . 04 . 55 . 05 . 56 . 06 . 57 . 07 . 5
G � l n � G ��������
G�����
��� , G
�ln��
�������
Fig. 13: Average of the differences between each ensem-ble and true model after each data assimilation.
SPE 95750 13
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 - 3 . 0
0 . 1
3 . 3
6 . 4
9 . 5
Fig. 14: log-permeability field, truth
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 - 3 . 0
0 . 1
3 . 3
6 . 4
9 . 5
Fig. 15: Average log-permeability after 540 days data as-similation.
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 - 3 . 0
0 . 1
3 . 3
6 . 4
9 . 5
Fig. 16: Average log-permeability after 5580 days dataassimilation.
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 - 3 . 0
0 . 1
3 . 3
6 . 4
9 . 5
Fig. 17: Average log-permeability after 7290 days dataassimilation.
14 SPE 95750
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 0 . 0 6
0 . 1 3
0 . 2 0
0 . 2 8
0 . 3 5
Fig. 18: porosity field, truth
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 0 . 0 6
0 . 1 3
0 . 2 0
0 . 2 8
0 . 3 5
Fig. 19: Average porosity after 7290 days data assimila-tion.
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 0 . 2 0
0 . 3 8
0 . 5 5
0 . 7 3
0 . 9 0
Fig. 20: Water saturation profile after 7290 days, truth.
5 1 0 1 5 2 0
5
1 0
1 5
2 0
2 5
3 0 0 . 2 0
0 . 3 8
0 . 5 5
0 . 7 2
0 . 9 0
Fig. 21: Average water saturation profile after 7290 daysdata assimilation.
SPE 95750 15
7 5 0 0 8 0 0 0 8 5 0 0 9 0 0 0 9 5 0 0 1 0 0 0 02 . 0 x 1 0 6
4 . 0 x 1 0 6
6 . 0 x 1 0 6
8 . 0 x 1 0 6
1 . 0 x 1 0 7
1 . 2 x 1 0 7
1 . 4 x 1 0 7
Cumu
lative
Oil P
rodu
ction
, STB
T i m e , d a y
Fig. 22: Cumulative oil production prediction for initial ensembles.
7 5 0 0 8 0 0 0 8 5 0 0 9 0 0 0 9 5 0 0 1 0 0 0 02 . 0 x 1 0 6
4 . 0 x 1 0 6
6 . 0 x 1 0 6
8 . 0 x 1 0 6
1 . 0 x 1 0 7
1 . 2 x 1 0 7
1 . 4 x 1 0 7
Cumu
lative
Oil P
rodu
ction
, STB
T i m e , d a y
Fig. 23: Cumulative oil production prediction after 5400days data assimilation.
7 5 0 0 8 0 0 0 8 5 0 0 9 0 0 0 9 5 0 0 1 0 0 0 02 . 0 x 1 0 6
4 . 0 x 1 0 6
6 . 0 x 1 0 6
8 . 0 x 1 0 6
1 . 0 x 1 0 7
1 . 2 x 1 0 7
1 . 4 x 1 0 7
Cumu
lative
Oil P
rodu
ction
, STB
T i m e , d a y
Fig. 24: Cumulative oil production prediction after 7290days data assimilation.
16 SPE 95750
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 0
6 0 0 0
7 0 0 0 T r u e M o d e l C e n t r a l M o d e l O b s e r v e d D a t a E n s e m b l e s
WBHP
, psi
T i m e , d a y
Fig. 25: Well bottom hole pressure for INJ-1(PRO-1),data assimilation for 5400 days, prediction to10000 days.
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 0
6 0 0 0
7 0 0 0 T r u e M o d e l C e n t r a l M o d e l O b s e r v e d D a t a E n s e m b l e s
WBHP
, psi
T i m e , d a y
Fig. 26: Well bottom hole pressure for INJ-1(PRO-1),data assimilation for 7290 days, prediction to10000 days.
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00
5 0 01 0 0 01 5 0 02 0 0 02 5 0 03 0 0 03 5 0 04 0 0 0
T r u e M o d e l C e n t r a l M o d e l O b s e r v e d D a t a E n s e m b l e s
WBHP
, psi
T i m e , d a y
Fig. 27: Well bottom hole pressure for PRO-3, data as-similation for 5400 days, prediction to 7290days.
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00
5 0 01 0 0 01 5 0 02 0 0 02 5 0 03 0 0 03 5 0 04 0 0 0
T r u e M o d e l C e n t r a l M o d e l O b s e r v e d D a t a E n s e m b l e s
WBHP
, psi
T i m e , d a y
Fig. 28: Well bottom hole pressure for PRO-3, data as-similation for 7290 days.
SPE 95750 17
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00 . 00 . 40 . 81 . 21 . 62 . 02 . 42 . 83 . 23 . 64 . 04 . 4
T r u e M o d e l C e n t r a l M o d e l O b s e r v e d D a t a E n s e m b l e s
WGOR
, MSC
F/STB
T i m e , d a y
Fig. 29: Production GOR for PRO-3, data assimilationfor 5400 days, prediction to 10000 days.
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00 . 00 . 40 . 81 . 21 . 62 . 02 . 42 . 83 . 23 . 64 . 04 . 4
T r u e M o d e l C e n t r a l M o d e l O b s e r v e d D a t a E n s e m b l e s
WGOR
, MSC
F/STB
T i m e , d a y
Fig. 30: Production GOR for PRO-3, data assimilationfor 7290 days, prediction to 10000 days.
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 00 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
T r u e M o d e l C e n t r a l M o d e l E n s e m b l e s
Water
Cut
T i m e , d a y
Fig. 31: Water cut for PRO-5, data assimilation for 7290 days.
18 SPE 95750
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0
T r u e M o d e l C e n t r a l M o d e l E n s e m b l e s
Water
Cut
T i m e , d a y
Fig. 32: Water cut for PRO-2, data assimilation for 5400days, prediction to 10000 days.
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 00 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0
T r u e M o d e l C e n t r a l M o d e l E n s e m b l e s
Water
Cut
T i m e , d a y
Fig. 33: Water cut for PRO-2, data assimilation for 7290days, prediction to 10000 days.