SPE 109975 Estimation of Initial Fluid Contacts by ... · for uncertainty in the depths of the initial ﬂuid contacts and provide estimates of these depths in addition to the tradi-tional

SPE 109975

Estimation of Initial Fluid Contacts by Assimilation of Production DataWith EnKF

Kristian Thulin, CIPR, U. of Bergen, Gaoming Li, SPE, U. of Tulsa, Sigurd Ivar Aanonsen, SPE, CIPR, U. of Bergen, AlbertC. Reynolds, SPE, U. of Tulsa

Copyright 2007 Society of Petroleum Engineers

This paper was prepared for presentation at the 2007 SPE Annual Technical Conferenceand Exhibition held in Anaheim, California, U.S.A., 11–14 November 2007.

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AbstractMuch recent work on automatic history matching and dataassimilation has focused on the adjustment of simulator grid-block permeabilities and porosities. Here, we show that whenproduction data are assimilated into reservoir models withthe ensemble Kalman filter, it is relatively easy to accountfor uncertainty in the depths of the initial fluid contacts andprovide estimates of these depths in addition to the tradi-tional estimates of rock properties fields. The contact depthsstrongly affect the initial oil in place and cumulative oil pro-duction. We demonstrate that if one uses fixed, but incorrectfluid contacts when assimilating data with EnKF, reasonablematches of production data are obtained, but future perfor-mance predictions are inaccurate and severely biased. Con-sidering these same inaccurate contact depths as the meansof probability density functions for the two depths and in-cluding both the contact depths and rock property fields inthe EnKF state vector, we obtain improved performance pre-dictions compared with the case where incorrect depths areassumed to be correct. With uncertain initial fluid contacts,the estimates of the location (depths) of the contacts obtainedby matching production data are more accurate than the priorestimates, but unfortunately, do not always provide an im-proved estimate of the thickness of the oil column. However,this is reflected in a larger estimated uncertainty in the pre-dictions.

In the reservoir engineering community, there exists someuncertainty about whether performance prediction runs witha reservoir simulator should be made by predicting forwardfrom the end of data assimilation or by rerunning the reser-voir simulator from time zero using the final ensemble ofreservoir parameters obtained at the final data assimilationstep. We show that if the model error is negligible and therelation between data and the combined state vector is linear,then both procedures give the same predictions. We demon-strate, that for the nonlinear examples considered here thetwo procedures give reasonably consistent results, although

rerunning from time zero tends to give slightly larger esti-mates of uncertainty in the predictions.

IntroductionThe ensemble Kalman filter (EnKF), introduced by Evensen1,has been used extensively for forecasts of dynamic variablesin meteorological and oceanographic systems. Since its in-troduction into the petroleum engineering literature as a methodfor real-time assisted history matching by Nævdal et al.2,3,4

for estimation or stochastic simulation of both reservoir mo-del parameters and reservoir simulator primary variables, ithas been a focus of much research activity in the reservoirengineering literature5,6,7,8,9,10,11,12. A discussion of the com-bined parameter and state estimation problem can be foundin Evensen13. In the reservoir engineering literature, EnKFhas been primarily used to estimate or stochastically simu-late gridblock permeabilities and porosities. However, it canconceptually be extended to include other parameters suchas depths of fluid contacts and fault transmissibilities14 or thelocation of boundaries between facies7.

As noted above, most EnKF history matching implemen-tations have focused on the estimation or simulation of grid-block permeabilities and porosities, properties that are easyto update sequentially in reservoir simulators. In this case,all reservoir parameters not included in the estimation, andthat in reality are not perfectly known, are assumed to becorrect during the history matching phase. This can result intuning of the wrong parameters to compensate for the modelerror introduced by an incorrect parameter. In particular, thedepths of the initial fluid contacts are often not known accu-rately but have a large impact on oil in place and the produc-tion of hydrocarbons as well as water.

Very recently, Evensen14 presented the results of a fieldhistory match in which he was able to use EnKF to success-fully match data by updating gridblock permeabilities andporosities, fault transmissibilities and the locations of initialfluid contacts together with the dynamic variables. Our ob-jective, here, is to investigate the reliability of EnKF for up-dating both the depths of the initial fluid contacts and therock property fields for a reservoir model where the truth isknown exactly. Specifically, we apply EnKF to the PUNQ-S3 model15 to update both contact depths and simulator grid-block rock properties given prior Gaussian probability den-sity functions where the best prior estimates of contact depths(the prior means) differ significantly from the depths of thetrue case. We compare reservoir performance predictions ob-tained with this process with those obtained by updating onlyrock property fields with (i) the contact depths set equal totheir prior means, and (ii) the case where the location of fluid

2 SPE 109975

contacts are set equal to the true values during data assimila-tion.

The outline of the paper is as follows. A short descriptionof the EnKF methodology is given in the next section fol-lowed by an explanation of a concept of consistency for thecombined parameter and dynamic state estimation. A briefdescription of the PUNQ-S3 model is given and the two ex-amples we consider are discussed. In the first example, theprior pdfs for the depths of the contacts have means whichindicate an oil column thickness much smaller than the truth.In the second example, the prior pdfs both have mean depthslarger than the truth, but the thickness of the oil column com-puted as the difference between the means of the contacts isthe same as for the truth case. Finally we give some conclu-sions.

Estimating Initial Fluid Contacts With the Ensemble Kal-man FilterThe ensemble Kalman filter is a sequential Monte Carlo met-hod developed for data assimilation for non-linear dynamicalsystems. It consists of a two step process at each data as-similation time, a prediction or forecast based on the stateestimate from the previous data assimilation step, and an up-date of the state variables by assimilating or matching theobserved data available at the current assimilation time step.

When using EnKF to solve a combined parameter and stateestimation problem, the state vector is augmented with theparameters to a combined vector consisting of a vector ofmodel parameters denoted here by α and a vector of dynamicvariables denoted here by xk, i.e., the combined state vectoris given by

ψk =[

xkα

], . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where k denotes the data assimilation time. For simplicity,we assume here that observed data is assimilated at everytime step so xk is the dynamic state at time tk.

Consider the non-linear model

ψk = g(ψk−1)+ εmk ε

mk ∼ N(0,Cεm

k) . . . . . . . . . . (2)

dk = Hψk + εdk ε

dk ∼ N(0,C

εdk), . . . . . . . . . . . . (3)

where g represents the dynamic system for propagating themodel from time tk−1 to time tk, εm

k is the assumed Gaussianmodel error, dk are data measurements, and εd

k is the mea-surement error, which is also assumed to be Gaussian.

The ensemble Kalman filter uses an ensemble of realiza-tions,

{ψak,i}

Nei=1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where at the k’th assimilation step, each ensemble member isa sample from the estimated posterior distribution f (ψk|dk:1).Here dk:1 denotes all the observed data up to and includingtime tk, d1,d2, . . . ,dk , and Ne is number of ensemble mem-bers. The superscript a in (4) denotes the analyzed or updatedstate, as opposed to the forecast or predicted state which isdenoted with a superscript f .

At the forecast step, each of the ensemble members is prop-agated in time, adding sampled model noise so that

ψfk,i = g(ψa

k−1,i)+ εmk,i, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

for i = 1,2, · · ·Ne. In the ensemble Kalman filter, the co-variance matrix at each time step is estimated from the Neensemble members by the equation

C fψk =

1Ne −1

Ne

∑i=1

(ψ

fk,i −ψ

fk

)(ψ

fk,i −ψ

fk

)T, . . . . . (6)

where

ψfk =

1Ne

Ne

∑i=1

ψfk,i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

At the analysis step, each ensemble member is updated, us-ing different perturbed data for each ensemble member, withthe regular Kalman filter update equations16,13 to obtain

ψak,i = ψ

fk,i +Kk

(dk,i −Hψ

fk,i

), . . . . . . . . . . . . . . . . . . . (8)

Kk = C fψk HT

(HC f

ψk HT +Cεd

k

)−1, . . . . . . . . . . . . . . . . (9)

where dk,i = dk + εdk,i is the perturbed data17, and the covari-

ance matrix C fψk is estimated from the forecast ensemble. For

a linear forward model with Gaussian statistics, the EnKFsamples exactly, as the ensemble size approaches infinity.

After the last assimilation step, predictions with uncer-tainty may normally be generated by simply running the sim-ulation from the last data assimilation time for all the modelsin the ensemble. Since there is no effective way to changefluid contacts during the middle of a reservoir simulation run,the estimated fluid contact depths will not affect the predic-tions forward from the last data assimilation step and do notaffect the updated rock property fields in this case. Thus,even though they are updated during each assimilation step,they cannot be used explicitly until we return to time zero andrun all predictions forward. However, the saturation changesduring the assimilation run will reflect the movements of saythe oil column, and in the examples shown, the predictionsobtained by just running the simulations forward are not sub-stantially different from those obtained by rerunning the sim-ulator from time zero using the updated ensemble of fluidcontacts and properties.

In the next section we address the consistency betweenpredictions obtained by simulating forward from the last dataassimilation time, and by rerunning from time zero for a lin-ear model.

ConsistencyWhen using the Kalman filter or the ensemble Kalman filterto solve the combined parameter and state estimation prob-lem, we would like the method to be consistent. By this, wemean that the state estimates, x f

k or xak , obtained using the

filter are the same as what we would obtain if we ran themodel with the corresponding parameter estimates, α

fk or αa

krespectively, from time zero with an initial state x0. In thiscase, with uncertainty initial fluid contacts, and thus in theinitial saturations and pressures, we also need to establishconsistency where there is uncertainty in the initial state x0.We show consistency for a general linear model as long asexplicit model errors can be neglected, i.e., if the uncertaintyin the model due to uncertainty in the model parameters ismuch larger than the uncertainty due to model errors.

Consider the general linear model

xk = Fkα+Akxk−1 + εmk , . . . . . . . . . . . . . . . . . . . . . . . . (10)

SPE 109975 3

dk = Hα,kα+Hxk xk + εdk , . . . . . . . . . . . . . . . . . . . . . . . (11)

and introduce the recursive notation

Fk = Fk +AkFk−1 F0 = 0 . . . . . . . . . . . . . . . . . . . (12)

Ak = AkAk−1 A0 = I . . . . . . . . . . . . . . . . . . . . . (13)

εmk = ε

mk +Akε

mk−1 ε

m0 = 0. . . . . . . . . . . . . . . . . . . . . (14)

With this notation, Eq. 10 becomes

xk = Fkα+ Akx0 + εmk . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

Because the expectation of the model error is zero, consis-tency, as defined above, requires that the estimates at eachtime step satisfy

x fk = Fkα

fk + Akx f

0,k, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

xak = Fkα

ak + Akxa

0,k, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

where e.g. xa0,k is the estimate of x0 after assimilating data

up to dk. Eq. 16 follows immediately from Eq. 17 from theprevious time step, so we need to show that Eq. 17 is true forall k. Since this is a linear model, consistency will be shownusing the classical Kalman filter.

Consistency will now be proved by mathematical induc-tion. The case for k = 0 is straightforward, so we make theinduction assumption that after assimilating observed datadk−1 we have consistency, i.e.,

xak−1 = Fk−1α

ak−1 + Ak−1xa

0,k−1. . . . . . . . . . . . . . . . . . (18)

After the next forecast (prediction) in the filter, the estimatesare

αfk = α

ak−1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

x f0,k = xa

0,k−1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

x fk = Fkα

fk + Akx f

0,k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

The analysis update is now xak

αak

xa0,k

=

x fk

αfk

x f0,k

+

C fxk,dk

C fαk,dk

C fx0,dk

(C f

dk,dk+C

εdk

)−1(

dk −H[

x fk

αfk

]), .. (22)

where

H = [ Hxk Hα,k ] , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

and

H[

x fk

αfk

]= Hxk x f

k +Hα,kαfk . . . . . . . . . . . . . . . . . . . . . . (24)

represents predicted data at tk. From Eqs. 10 and 21, wenotice that

xk − x fk = Fk

(α−α

fk

)+ Ak

(x0 − x f

0,k

)+ ε

mk , . . (25)

which allows us to write the conditioned cross-covariancematrix between the dynamic variables and the data as

C fxk,dk

= E[(

xk − x fk

)(dk −Hψk)

T ∣∣ dk−1:1

], ......... (26)

= E[(

Fk

(α−α

fk

)+ Ak

(x0 − x f

0,k

)+ ε

mk

)·

(dk −Hψk)T ∣∣ dk−1:1

], ......................... (27)

= FkE[(

α−αfk

)(dk −Hψk)

T ∣∣ dk−1:1

]+ AkE

[(x0 − x f

0,k

)(dk −Hψk)

T ∣∣ dk−1:1

]+E

[ε

mk (dk −Hψk)

T ∣∣ dk−1:1

], ............... (28)

≈ FkCfαk,dk

+ AkCfx0,dk

, ................................... (29)

where the final approximation in the last equation assumesthat the error in the model noise is negligible compared tothe error though the unknown parameters (x0 and α).

Substituting the final expression in Eq. 29 into Eq. 22 gives,after some algebraic manipulations and using Eq. 21,

xak = x f

k + Fk

(α

ak −α

fk

)+ Ak

(xa

0,k − x f0,k

)= Fkα

fk + Akx f

0,k + Fk

(α

ak −α

fk

)+ Ak

(xa

0,k − x f0,k

)= Fkα

ak + Akxa

0,k. ............................................... (30)

We have assumed Eq. 18 and deduced Eq. 30, and thus provedthe consistency in Eqs. 16 and 17 by induction. If in ourmodel, we have model noise we have to assume that themodel noise is negligible compared to the mismatch due tothe unknown parameters in order to have consistency.

PUNQ-S3The reservoir model used is the well known small syntheticreservoir engineering model, the PUNQ-S3 test case15. Thestructure is dome shaped. The top structure of the field isshown in Fig. 1. There are six producers and no injectors, thereservoir is supported by a fairly strong aquifer on the northand west side. A fault bounds the reservoir on the south andeast sides. This model is based on a real field and was set upas a test case to evaluate methods for assessing uncertaintyin reservoir description and performance predictions. Gao etal.11 found that for this problem, EnKF and randomized max-imum likelihood both gave similar and reasonable character-izations of the uncertainty in future performance predictions.More details on the PUNQ-S3 model can be found in Floriset al.18 and on the PUNQ-S3 web page.15

ExamplesTo evaluate the effect of unknown contacts, we first assim-ilate data using the standard EnKF where the fluid contactsare fixed at their true values. We use an ensemble consist-ing of 90 realizations where the initial ensemble is generatedfrom the geological model derived by Gao et al.10. Well-bore pressures, GOR and watercut data over a 11 year period(4,023 days) are used to update gridblock porosities and hor-izontal and vertical log-permeabilities. All results are plottedin field units with time in days. To evaluate the consistency ofthe method for this non-linear model, we have also done pre-dictions by running the simulator from time zero using the

4 SPE 109975

Figure 1—The top structure map for the PUNQ-S3

TABLE 1—INPUT DATA FOR FLUID CONTACTS

GOC stdGOC OWC stdOWCTruth 7726.38 - 7857.61 -Example A 7746.38 30 7837.61 30Example B 7756.38 30 7887.61 30

ensemble of rock property fields obtained at the last EnKFassimilation step. While more computationally expensive,running predictions from time zero makes it possible to ex-plicitly incorporate the updated contact depths. It has alsothe advantage that it guarantees preservation of material bal-ance for each ensemble member and that nonphysical valuesof reservoir simulation primary variables are avoided.

We consider two general examples. The main differencebetween the examples is in the prior mean depth for the loca-tion of the fluid contacts. In Example A, the means are suchthat the mean depth of the water-oil contact (WOC) is 20 feethigher than the true depth and the mean depth of the gas-oilcontact (GOC) is 20 feet lower than the true depth. The neteffect is that the best prior estimate of the oil column thick-ness is 40 feet less than the true thickness. In Example B, themean depth of both contacts is 30 feet lower than the truthso the best prior estimate of the thickness of the oil columnis equal to the true thickness. We assume a prior Gaussiandistribution for the depths of the contacts. The pertinent in-formation on the prior models is given in Table 1 where stddenotes the standard derivation and the other entries refer tothe mean depths.For each example, we consider four cases, which are labelled(a) through (d) in the figures and defined below.

• (a) The depths of the fluid contacts are fixed at the truevalues. Data for 11 years are assimilated with EnKF toupdate only log-permeability and porosity fields.

• (b) Contact depths are fixed at the incorrect valuesgiven by the prior means in Table 1, and only log-permeability and porosity fields are updated by dataassimilation with EnKF.

• (c) Here, we include the depths of the contacts along

0 500 1000 1500 2000 2500 3000 3500 4000−7950

−7900

−7850

−7800

−7750

−7700

−7650

Figure 2—The updates of the mean and the uncertainty of theinitial fluid contacts during the assimilation. The dashed linesare two standard deviations from the mean estimates. The truevalues are shown by the dotted lines.

TABLE 2—ESTIMATED FLUID CONTACTSEXAMPLE A

meanGOC stdGOC meanOWC stdOWC7717.2 7.1 7868.7 6.5

with the log-permeabilities and porosities as model pa-rameters. The means and standard derivations of thedepths for the prior Gaussian distribution of the twoinitial fluid contacts are given in the Table 1. Usingthese distributions, we generate a set of 90 realizationsof each fluid contact depth and combine them withthe 90 initial ensemble members for the rock propertyfields. The initial ensemble of the rock property fieldsis the same in all cases. Note that the estimated con-tacts are not explicitly used for predictions, except forthe case when the simulator is rerun from time zerousing the final updated ensemble members.

• (d) As in case (c), the initial contacts and rock propertyfields are updated by assimilating the observed dataduring the first 11 years with EnKF. Then, we returnto time zero and use the ensemble of updated contactdepths together with the initial (not updated) ensem-ble of rock property fields as the initial ensemble fora second EnKF data assimilation run. In the secondrun, we update only the gridblock log-permeabilitiesand porosities, not the contacts. In this case the “rerunpredictions”, starting from time zero, are based on a setof realizations obtained by using the updated ensembleof rock property fields from the second data assimila-tion run together with the updated contact depths fromthe first run. Although the second assimilation is con-sistent with itself, this two-step procedure will not beconsistent as defined above for a linear case, since theinitial correlations between contact depths and porosi-ties and permeabilities depend on the data for the sec-ond assimilation. However, such a sequential updatemay still give the best results in practice for a non-linear case.

Example A. Fig. 2 shows how the estimate and the uncer-tainty in the fluid contacts are updated during the EnKF dataassimilation for cases (c) and (d). Note that the estimateddepths mean moves towards the truth giving a thicker oil col-umn and the uncertainty band narrows as more data are as-

SPE 109975 5

similated, but unfortunately, overshooting occurs. The finalestimates for the mean and standard deviation of the con-tact depths are given in Table 2. It is seen that EnKF ac-tually over corrected as the estimated depth of the WOC is11 feet greater than the true depth and the estimated depthof the GOC is 9 feet smaller than the truth. Subtractingthe estimated mean depth of the GOC from the estimatedmean depth of the WOC gives an estimated oil column heightof 151.5 feet as opposed the true height which is equal to131.23. Thus, the prior means result in an estimated oil col-umn thickness which is 40 feet less than the true thickness,whereas EnKF over estimates the thickness of the oil columnby 20.3 feet.

Figs. 3 through 13 show the predictions for cases (a), (b)(c) and (d).

Figs. 3, 4 and 5, respectively, show ensemble predictionsof bottom-hole pressure, producing GOR and water cut fromone of the producers (PRO-11) compared to the data and re-sults obtained by running the true model. The data are ob-tained by adding Gaussian noise to the results from the truemodel. The ensemble predictions are obtained by plottingthe forecasts steps during the data assimilation process foreach member of the ensemble. Thus the future predictionsare obtained by simply running the simulation forward fromthe last data assimilation time for another 5.5 years (1,991days).

As noted previously, in case (b) we adjust only rock prop-erty fields by assimilating production data with contacts depthsfixed equal to their prior means (Table 1) so that the thicknessof the oil column is fixed and has a value 40 feet less than thethickness of the true reservoir associated with the observeddata. The results of Figs. 3(b), 4(b) and 5(b) indicate thateven though we used fixed incorrect depths of fluid contacts,we are able to match most of the historical data fairly well,but we see that predictions are significantly worse than in theother cases. The trend in predicted water cut is incorrect dur-ing later prediction period (Fig. 5(b)), and there is a moreclear bias in the predicted GOR (Fig. 4(b)) than in the othercases.

It should be noted that zero water cut measurements atearly times were not used in the analysis for this well, re-sulting in too early predicted water break through. This isadjusted for at later times when observed water cut data areassimilated. Although predictions are shown for only onewell, the results are similar for the other wells. In general,the predictions of wellbore pressures and gas-oil ratios arefairly good, while the water cut predictions are of varyingquality, and with a large uncertainty. This may be becausethe estimated water-oil contact is too deep.

Fig. 6 shows predictions of water cut obtained by rerun-ning from time zero using the final analyzed ensemble ofparameters. Note that these predictions are reasonably con-sistent with those obtained by predicting from the last assim-ilation step (Fig. 5). However, the spread in the ensemblepredictions is slightly larger for the rerun in cases (c) and(d).

Corresponding plots for field cumulative oil, water and gasproduction for the four cases are shown in Figs. 7 through 12.The spread in cumulative oil production is not very large, andquite similar for all cases. Case (b) over predicts water pro-duction as expected because the depth of the WOC is muchshallower, and hence closer to the lower completions of theproducing wells than in the true reservoir. It seems that tomatch initial pressure drawdown, the EnKF over-estimatespore volume in all cases. At late times, this results in an

TABLE 3—FLUID VOLUMES EXAMPLE A

Pore volume STOIIP WIIP Free GIIPMMRb MMSTB MMSTB 109SCF

True 220.1 109.3 78.7 12.9Prior mean 220.1 76.7 107.0 26.4Estimated mean (a) 241.3 119.0 89.7 11.0Estimated mean (b) 245.0 88.5 116.6 28.0Estimated mean (c) 235.7 128.1 75.8 7.7Estimated mean (d) 225.0 124.1 69.8 7.9

TABLE 4—ESTIMATED FLUID CONTACTSEXAMPLE B

meanGOC stdGOC meanOWC stdOWC7721.4 7.2 7890.1 14.1

average pressure that is too high (Fig. 13), which results ina slight over-prediction of cumulative oil production. How-ever, for case (b) this is compensated by the wrong contacts,which results in a too high water and gas production. Incase (b), the prediction of cumulative oil is good, but for thewrong reason. True, initial, and estimated mean fluid vol-umes are listed in Table 3. Although over compensating, itis clearly seen that including the contacts as parameters im-proves the estimate of initial oil in place. In case (b), theSTOIIP is far too low, and quite close to the initial mean,while most of the pore volume adjustments are incorrectlyimposed in the water filled parts of the reservoir.

With respect to gas (Figs. 11, 12), the predictions fromcase (b) with the contact depths fixed at incorrect values areclearly biased, significantly over-predicting cumulative gasproduction because the gas-oil contact is much closer to thewell than in the truth case.

The apparently strange behavior of the water and gas pre-dictions, seen in Figs. 9(c) and 11(c), where there is a largeuncertainty initially, which do not increase further with time,is due to the initial uncertainty in contacts. Since the wellsare controlled by oil the rate, the variations in contacts depthsresult in a large variation in early water production, producedgas-oil ratios and bottom-hole pressures within the ensem-ble, and thus a large uncertainty in water and gas produc-tion. Later, when more data have been assimilated, this un-certainty is reduced, and the spread in cumulative productiondoes not increase. Thus, when comparing the “continuous”predictions of cumulative production (Figs. 7, 9, 11) with thepredictions from time zero (Figs. 8, 10, 12), only the valuesafter the last assimilation steps are relevant.

Example B. In example B, we also assume the prior meansof the contact depths are not equal to the true depths, but thedifference between the means gives the true height of the oilcolumn. According to the prior model shown in the table,the prior mean of each initial fluid contact depth is 30 feetgreater than the true depth. Fig. 14 shows the developmentof the fluid contact estimates during assimilation. The finalestimates for the contact depths are given in Table 4. Re-call that the true depth of the WOC is 7857.6 ft and the truedepth of the GOC is 7726.4 ft. Thus, like in Example A, ourestimate of the depth of the initial GOC is reasonably good,but our estimate of the depth of the WOC is too low. In fact,the mean value of WOC is hardly changed at all during theassimilation. The difference in the means gives an estimatedoil column thickness of 168.7 feet, about 37.5 feet greaterthan the true thickness.

The overall results of Example B are quite similar to Ex-ample A, and thus we show only plots of cumulative produc-

6 SPE 109975

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Figure 3—Ensemble predictions (EnKF forecasts) of BHP forproducer PRO-11 (blue lines), compared with true values (redline) and assimilated data (black crosses). Example A.

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Figure 4—Ensemble predictions (EnKF forecasts) of GOR forproducer PRO-11 (blue lines), compared with true values (redline) and assimilated data (black crosses). Example A.

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Figure 5—Ensemble predictions (EnKF forecasts) of water cutfor producer PRO-11 (blue lines), compared with true values(red line) and assimilated data (black crosses). Example A.

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Figure 6—Predictions of water cut for producer PRO-11 fromtime zero using the final analyzed ensemble of parameters (bluelines), compared with true values (red line) and assimilated data(black crosses). Example A.

8 SPE 109975

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Figure 7—Ensemble predictions (EnKF forecasts) of field cu-mulative oil production (blue lines), compared with true values(red line). Example A.

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Figure 8—Predictions of field cumulative production oil fromtime zero using the final analyzed ensemble of parameters (bluelines), compared with true values (red line). Example A.

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Figure 10—Predictions of field cumulative water productionfrom time zero using the final analyzed ensemble of parame-ters (blue lines), compared with true values (red line). ExampleA.

10 SPE 109975

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Figure 11—Ensemble predictions (EnKF forecasts) of field cu-mulative gas production (blue lines), compared with true values(red line). Example A.

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Figure 12—Predictions of field cumulative gas production fromtime zero using the final analyzed ensemble of parameters (bluelines), compared with true values (red line). Example A.

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Figure 14—The updates of the mean and the uncertainty of theinitial fluid contacts during the assimilation. The dashed linesare two standard deviations from the mean estimates. The truevalues are shown by the dotted lines.

tion of oil gas and water (Figs. 15 through 20). Apparently,the data as defined here does not provide sufficient informa-tion to condition the water-oil contact. However, this is re-flected in a larger uncertainty in the predictions, and althoughthe estimates of initial contacts are not very good, the ensem-ble predictions in case (c) and (d) cover the true cumulativewater and gas production in both examples, while the predic-tions in case (b), with assumed known, but incorrect contacts,are clearly biased with a low estimated uncertainty.

Comments and ConclusionsA methodology and motivation for including depths of theinitial fluid contacts into the ensemble Kalman filter has beenpresented. The initial fluid contacts might be poorly knownin many cases, and we have shown that it is critical to includethe contact depths as parameters when assimilating data withEnKF. Failure to do so may lead to erroneous and biased fu-ture performance predictions. With contact depths fixed atincorrect values, we obtained reasonable data matches withthe wrong model, but future predictions were biased.

Reasonable performance predictions were obtained whenupdating the contacts along with the rock property fields us-ing EnKF. Two methods were given for including uncertaintyin the contacts, and from a cursory examination of all predic-tions on a well by well basis, it not clear which one is best.Further investigations are needed.

We show that the Kalman filter and ensemble Kalman filterfor combined state and parameter estimation are consistentfor a linear model. That is, the state estimates obtained usingthe filter are the same as what we would obtain if we ran themodel with the corresponding parameter estimates from timezero. We demonstrate that the EnKF is reasonably consistentalso when updating rock properties and/or fluid contacts forthe PUNQ-S3 case.

AcknowledgmentsThis work was supported partly by the member companies ofthe The University of Tulsa Petroleum Reservoir Exploita-tion Projects, TUPREP, and partly by the Norwegian Re-search Council, PETROMAKS programme. SchlumbergerInformation Services is acknowledged for providing reser-voir simulator software licenses. We thank Hans Julius Skaugfor helpful discussions on the consistency section.

12 SPE 109975

References1. Evensen, G.: “Sequential data assimilation with a nonlin-

ear quasi-geostrophic model using Monte Carlo methodsto forecast error statistics,” J. Geophys. Res. (1994) 99,10,143.

2. Nævdal, G., Mannseth, T. and Vefring, E.: “Near-WellReservoir Monitoring Through Ensemble Kalman Filter,”SPE/DOE Improved Oil Recovery Symposium, Tulsa, Ok-lahoma (April 2002) SPE 75235.

3. Nævdal, G., Johnsen, L., Aanonsen, S. and Vefring, E.:“Reservoir Monitoring and Continuous Model UpdatingUsing Ensemble Kalman Filter,” SPE Annual TechnicalConference and Exhibition, Denver, Colorado, USA (5 –8 October 2003) SPE 84372.

4. Nævdal, G., Johnsen, L., Aanonsen, S. and Vefring, E.:“Reservoir Monitoring and Continuous Model UpdatingUsing Ensemble Kalman Filter,” SPE Journal (March2005) .

5. Gu, Y. and Oliver, D.: “The ensemble Kalman filterfor continuous updating of reservoir simulation models,”Submitted Computational Geosciences (January 2004).

6. Skjervheim, J. et al.: “Incorporating 4D Seismic Datain reservoir Simulation Models Using Ensemble KalmanFilter,” SPE Annual Technical Conference and Exhibition,Dallas, Texas (9-12 October 2005) SPE95789.

7. Liu, N. and Oliver, D.: “Critical Evaluation of the Ensem-ble Kalman Filter on History Matching of Geologic Fa-cies,” SPE Reservoir Evaluation and Engineering (2005)8, 470.

8. Wen, X.H. and Chen, W.: “Real-Time Reservoir ModelUpdating Using Ensemble Kalman Filter,” SPE Reser-voir Simulation Symposium, Houston, Texas (31 January-2 February 2005) SPE92991.

9. Zafari, M. and Reynolds, A.: “Assessing the Uncertaintyin Reservoir Description and Performance PredictionsWith the Ensemble Kalman Filter,” SPE Annual TechnicalConference and Exhibition, Dallas, Texas (9-12 October2005) SPE95750.

10. Gao, G., Zafari, M. and Reynolds, A.: “Quantifying theUncertainty for the PUNQ-S3 Problem in a BayesianSetting With RML and EnKF,” SPE Reservoir Simula-tion Symposium, Houston, Texas (31 January-2 February2005) SPE93324.

11. Gao, G., Zafari, M. and Reynolds, A.: “Quantifying theUncertainty for the PUNQ-S3 Problem in a Bayesian Set-ting With RML and EnKF,” SPE Journal (2006) 11, 506.

12. Skjervheim, J. et al.: “Using The Ensemble Kalman Fil-ter With 4D Data to Estimate PRoperties and Lithologyof Reservoir Rocks,” Proceedings of the 10th EuropeanConference on the Mathematics of Oil Recovery, Amster-dam (4–7 September 2006) .

13. Evensen, G.: Data Assimilation: The Ensemble KalmanFilter, Springer, Berlin (2006).

14. Evensen, G. et al.: “Using the EnKF for Assisted HistoryMatching of a North Sea Reservoir Model,” SPE Reser-voir Simulation Symposium, Houston, Texas (26–28 Feb-ruary 2007) SPE106184.

15. http://www.nitg.tno.nl/punq/cases/punqs3.

16. Welch, G. and Bishop, G.: “An In-troduction to the Kalman Filter,”http://www.cs.unc.edu/ welch/kalman/kalmanIntro.html.

17. Burgers, G., van Leeuwen, P. and Evensen, G.: “On theAnalysis Scheme in the Ensemble Kalman Filter,” Mon.Weather Rev. (1998) 126, 1719.

18. Floris, F. et al.: “Methods for quantifying the uncertaintyin production forecasts: A comparative study,” PetroleumGeoscience (2001) SUPP, 87.

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Figure 15—Ensemble predictions (EnKF forecasts) of field cu-mulative oil production (blue lines), compared with true values(red line). Example B.

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Figure 16—Predictions of field cumulative oil production fromtime zero using the final analyzed ensemble of parameters (bluelines) compared to the true value (red line). Example B.

14 SPE 109975

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Figure 18—Predictions of field cumulative water productionfrom time zero using the final analyzed ensemble of parame-ters (blue lines) compared to the true value (red line). ExampleB.

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Figure 19—Ensemble predictions (EnKF forecasts) of field cu-mulative gas production (blue lines), compared with true value(red line). Example B.

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Figure 20—Predictions of field cumulative gas production fromtime zero using the final analyzed ensemble of parameters (bluelines) compared to the true value (red line). Example B.

SPE 109975 Estimation of Initial Fluid Contacts by ... · for uncertainty in the depths of the initial ﬂuid contacts and provide estimates of these depths in addition to the tradi-tional

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