4Den

g G

2006 Elsevier B.V. All rights reserved.

either the weighted least-square method or the Bayesianframework (Tarantola, 1987). If non-gradient basedminimization methods are used to minimize the

Journal of Petroleum Science and Engin

Corresponding author. Tel.: +1 281 544 4227.

E-mail addresses: Yannong.Dong@shell.com (Y. Dong),Keywords: Automatic history matching; Ensemble Kalman filter; Data assimilation; Time-lapse seismic

1. Introduction

The purpose of history matching is to adjust theparameters, such as permeability and porosity, in a

reservoir simulation model to enable the computedhistories, such as water production rate, oil productionrate, and time-lapse seismic data, to have reasonablecloseness with the observed histories (Reviewer 2,Comment 3). The matching process typically involvesthe minimization of an objective function derived fromAbstract

Automatic history matching of both production data and time-lapse seismic data to achieve reservoir characterization withreduced uncertainty has been extensively studied in recent years. Feasible applications, however, require either the adjoint methodor the gradient simulator method to compute the gradient/Hessian matrix of the objective function for the minimization algorithm.Both methods are computationally expensive when either the number of model parameters or the number of observed data is large.

In this paper, the ensemble Kalman filter (EnKF) is used to history match both production data and time-lapse seismicimpedance data. EnKF uses a set of reservoir models as input; continuously updates the models by assimilating observation datawhenever they are available; and outputs a number of history-matched models that are suitable for uncertainty analysis. SinceEnKF does not require the adjoint code, it is independent of reservoir simulators. A small synthetic case study was conducted,which shows the possibility of integrating both time-lapse seismic data and production data using the EnKF for reservoircharacterization. The observed data are matched very well, and the true model features are recovered. The estimated porosity fieldis better than the estimated permeability field because seismic data are directly sensitive to porosity but only indirectly sensitive topermeability. The improved initial member sampling algorithm helps to keep large variance space within ensemble members,ensuring stable filter behavior.a 200 North Dairy Ashford, WCK 3432, Houston, TX 77079, United Statesb 100 East Boyd Street, SEC T301, Norman, OK 73019, United States

Received 6 July 2005; received in revised form 26 March 2006; accepted 27 March 2006Sequential assimilation ofdescription using the

Yannong Dong a,, Yaqinyaqing.gu-1@ou.edu (Y. Gu), dsoliver@ou.edu (D.S. Oliver).1 Tel.: +1 405 325 1734.2 Tel.: +1 405 325 2921.

0920-4105/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2006.03.028seismic data for reservoirsemble Kalman filter

u b,1, Dean S. Oliver b,2

eering 53 (2006) 8399www.elsevier.com/locate/petrolobjective function, thousands of simulation runs,which evaluate the objective function at each iteration

Science and Engineering 53 (2006) 8399step, are usually needed to find the global minimizationpoint. When model size becomes large, it tends to beprohibitive to apply the non-gradient based minimiza-tion methods due to highly demanding simulation runs.If gradient based methods are chosen, for example thelimited memory BroydenFletcherGoldfarbShanno(LBFGS) method (Nocedal, 1980), the adjoint methodmay be employed to compute the gradient. The adjointequations, however, are highly dependent on thereservoir simulator, i.e., they require access to thesource code of the simulator. If switching to a differentreservoir simulator, it is necessary to revise or evenrewrite the adjoint equations, which would be very time-consuming.

On the other hand, with increasing deployment ofpermanent sensors to monitor pressure, temperature, orflow rate, the observed data become dense in timedomain. It is more important to keep the model up-to-date by assimilating the observed data as soon as theybecome available rather than simultaneously incorpo-rating all the data to provide a set of reservoir simulationmodels. Although the sequential data assimilation canalso be realized by minimizing the objective function, itis not computationally preferable because either theadjoint code is required or the time consumed for oneminimization process is too long. Therefore, aninvestigation of alternate automatic history matchingmethods is worthwhile.

Historically, the Kalman filter (Kalman, 1960) is themost widely used sequential data assimilation method.Although it represents the most widely applied anddemonstrably useful result to emerge from the statevariable approach of modern control theory (Sorenson,1985), it is inapplicable to non-linear systems. Since thereservoir simulation equations are highly non-linearwhen multiple phases co-exist, the Kalman filter isinappropriate for the typical automatic history matchingproblems in reservoir characterization. In 1994, Evensen(1994) introduced the ensemble Kalman filter (EnKF)that can be applied to the non-linear systems. The EnKFis independent of reservoir simulators and does notrequire the adjoint code. It outputs a set of estimatedmodels, which are suitable for uncertainty analysis.After its debut in 1994, the EnKF has achieved a numberof successful applications in meteorology (Andersonand Anderson, 1999; Evensen and van Leeuwen, 1996;Evensen, 1996, 2003; Hamill and Whitaker, 2001;Houtekamer and Mitchell, 1998, 2001). Recently,applications of the EnKF can also be found in hydrology(Reichle et al., 2002) and petroleum engineering (Gao etal., 2005; Gu and Oliver, 2005a,b; Liu and Oliver, 2005;

84 Y. Dong et al. / Journal of PetroleumNvdal et al., 2002, 2005; Wen and Chen, 2005).2. Ensemble Kalman filter

The basic methodology of the EnKF consists of theforecast step and the assimilation step. The forecast stepis to advance the state vectors from the current time(Reviewer 1, Comment 1) step to the next time step. Thestate vector contains the variables required to describethe system. The assimilation step is to adjust all thevariables in the state vectors to honor the observed data.In reservoir characterization, the forecast step isachieved by using a reservoir simulator. Therefore, thestate vector typically includes porosity, log permeability,pressure, and phase saturations at each simulationgridblock. Besides those model variables, the statevector also includes the computed data, such as thereservoir response and time-lapse seismic data. If thenumber of gridblocks is Nm and the number of computeddata is Nd, the dimension of the state vector y is 4Nm+Nd for a wateroil system with capillary pressureneglected (Reviewer 1, Comment 2). The forecast stepcan be written as

ypk; j f yuk1; j j 1; 2;: : :;Ne; 1where u denotes updated, p denotes predicted, fdenotes the reservoir simulator, k is the time step index, jis the ensemble member index, Ne is the number ofensemble members (Reviewer 1, Comment 3), yk1,j

u isthe jth updated state vector after the data assimilation atthe time step k1, and yk,jp is the predicted state vectorbased on all available information prior to the time stepk. Note that only dynamic variables, i.e., pressure andsaturation, and the computed data, change between k1and k. The static variables, i.e., porosity and permeabil-ity, remain unchanged. They are adjusted as well as thedynamic variables at the assimilation step.

Suppose that some measurements are obtained at thetime step k,

dobs;k dtrue;k ek ; 2

where dobs,k is the noise corrupted observation vectorfrom the true observation and k is the measurementnoise with dimension equal to Nd1, is usuallyassumed to be Gaussian. The covariance matrix of kis CD,k=E[kk

T], with dimension equal to NdNd, istypically assumed to be diagonal if only productionobservations are used. By assimilating the observation,the state vectors are updated using Eq. (3),

yuk; j ypk; j Ke;kdobs;k; jHkypk; j j 1; 2;: : :;Ne:

3

3. Integration of time-lapse seismic impedance data

The time-lapse seismic impedance data come fromthe time-lapse seismic, which typically consists of twoseismic surveys shot at the same location but at differenttimes (Reviewer 1, Comment 4). Each survey providesone seismic impedance data set. The definition of theseismic impedance is

4r

85Science and Engineering 53 (2006) 8399In Eq. (3), dobs,k,j is the perturbed observation vectorfor the jth ensemble member by adding random noise tothe observation dobs,k (Burgers et al., 1998). Hk=[0|I] isthe operator matrix with dimension equal to Nd(4Nm+Nd). The first 4Nm columns are filled withzeros. Hkyk,j

p extracts the computed data from the jthstate vector corresponding to its observations and doesnot involve any matrix multiplication. Ke,k is calledthe Kalman gain matrix with dimension equal to(4Nm+Nd)Nd. It is computed from the ensemblemembers using Eq. (4),

Ke;k Ppe;kHTk HkPpe;kHTk CD;k1; 4

where e denotes ensemble. In Eq. (4), Pe,k is thecovariance matrix computed from the ensemblemembers,

Ppe;k 1

Ne1

XNej1

ypk;j ypk

ypk;j ypkT

1Ne1

DY pk DY pk T ; 5

where y kp is the averaged state vector and Y k

p consistsof Ne column vectors, each of which is the differencebetween an ensemble state vector and the averaged statevector. Applying Eq. (5) to Eq. (4) gives

Ke;k 1Ne1DYpk HkDY pk T

1Ne1

HkDYpk HkDY pk T CD;k

1: 6

Eq. (6) is computationally efficient because onlyY k

p needs to be formed and stored instead of the wholecovariance matrix and HkY k

p does not involve matrixcomputation. The EnKF algorithm is outlined as

Step 1 Advance the ensemble state vectors in timeusing the reservoir simulator. Fill the vectorswith initial values if it is the first time step.

Step 2 At the time step k when the observations areavailable, stop advancing and fill the statevectors with the current model variables andcomputed data.

Step 3 Compute the averaged state vector using y pk 1Ne

PNej1 y

pk;j.

Step 4 Form Ykp and take entries from Y k

p usingHk, which is HkY k

p.Step 5 Compute the Kalman gain matrix using Eq. (6).Step 6 Update the ensemble state vectors using Eq. (3).Step 7 If it is the final step, STOP. Otherwise, go to

Y. Dong et al. / Journal of PetroleumStep 1.Z qVp qK 3 q2V 2s ; 7

where Vp is the P-wave velocity, is the body density, Kis the bulk modulus, and Vs is the shear wave velocity.Although Eq. (7) computes the P-wave impedance, theterm seismic impedance is exclusively used becausethe P-wave impedance is the only impedance used inthis work. To calculate the seismic impedance using thereservoir simulator output, such as pressure andsaturation at each gridblock, requires , K, and Vs,which are obtained by applying rock physics models(Dong and Oliver, 2005). In this synthetic study, boththe observed and the computed impedance data werecomputed using the Gassmann (1951) and Han et al.(1986) equations. Some previous work on historymatching of time-lapse seismic data, for exampleHuang et al. (1997), also used the Gassmann equations(Reviewer 2, Comment 1). Table 1 gives some rockproperties that are assumed to be known withoutuncertainties throughout the computation. The constantbulk moduli of oil and water were used because no gasphase exists in the synthetic wateroil problem so oiland water properties do not change much with pressure.However, to consider changes in bulk moduli of waterand oil only requires minor modification in rock physicsmodels used to calculate K and Vs in Eq. (7) and does notchange the EnKF framework (Reviewer 2, Comment 5).

Since an impedance datum at one gridblock iscomputed using only the pressure and saturation at thegridblock, the observation noise covariance matrix CD,kis still reasonably assumed to be diagonal. For real case

Table 1Parameters used for seismic impedance calculation

Parameter Value

Shaliness 0.2Sand modulus (Pa) 3.81010

Clay modulus (Pa) 2.121010

Density of solid (kg/m3) 2650Modulus of water (Pa) 2.39109Modulus of oil (Pa) 6.71108

rmeab

Science and Engineering 53 (2006) 8399studies, however, a non-diagonal CD,k probably needsto be constructed (Aanonsen et al., 2002, 2003).

In this work, the time-lapse seismic impedance datawere assimilated as data sets from two separate seismicsurveys. Because seismic impedance data are muchmore sensitive to porosity than to permeability (Dongand Oliver, 2002), the porosity field was recoveredbetter than the permeability field in the synthetic casestudy. For the impedance data from the first seismicsurvey, the EnKF assimilated them one by one, whichis: (1) extract variables from global state vectorscorresponding to one gridblock to form local ensemblestate vectors for the same block and assimilate theseismic impedance datum at that place; (2) adjustvariables belonging to the gridblock in the local state

Fig. 1. True log pe

86 Y. Dong et al. / Journal of Petroleumvectors and repeat Step 1 and Step 2 until all gridblocksare covered (Reviewer 2, Comment 6); (3) gather thelocally adjusted state vectors to form the updatedglobal state vectors and advance them in time. As thefirst seismic survey is at the fairly early stage ofproduction, the fluid flow field is barely constructed.Thus, the sensitivity correlation of seismic impedancedata across gridblocks is still quite weak, which makesthis one-by-one assimilation scheme suitable (Reviewer1, Comment 5). For the second seismic survey, theEnKF assimilated all the impedance data simultaneous-ly because our study showed that there was asensitivity connection across the whole reservoirmodel arising from the fluid flow.

4. Improved initial member sampling

The EnKF uses the sample mean and samplecovariance to approximate the population mean andpopulation covariance (Evensen, 1994, 2003), so to use alarge number of ensemble members is presumably betterthan to use a small number of ensemble members.Considering the computation expense, however, thenumber of ensemble members must be kept reasonable.Since the adjustments in the EnKF are within the spacespanned by the ensemble members, a small number ofmembers may not be able to provide large enoughadjustment freedom, resulting in filter divergence(Anderson, 2001; Anderson and Anderson, 1999; Even-sen, 1994, 2003, 2004; Houtekamer and Mitchell, 1998).

In this work, the improved initial member samplingalgorithm (Evensen, 2004) was used. The resampledmembers generated from the algorithm kept the samplestatistics introduced by the original large ensemble set.

ility and porosity.They provided larger adjustment space than that fromsame number of ensemble members by sampling the

Fig. 2. True water saturation distribution at day 200.

Fig. 3. Water rate (STB/day) at the injector before and after the EnKF (M128 group).

87Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399prior probability density function (hereafter pdf) andensured a stable filter behavior. The sampling algorithmis:

Step 1 Generate NT ensemble members by drawingsamples from the prior pdf and compute thecorresponding difference matrix Y.

Step 2 Perform the Singular Value Decomposition(hereafter SVD) of Y, Y=UVT, where Uis the left orthogonal matrix, V is the rightorthogonal matrix, and is a diagonal matrixwith the singular values of Y as its diagonalentries.

Step 3 Choose the firstNe largest singular values fromand store them into the matrix e. Correspond-ingly, the first Ne column vectors of U are chosenand saved into the matrix Ue.

Step 4 Generate an NeNe random matrix by samplingthe standard normal distribution N(0, 1).

Step 5 Perform the SVD of the random matrix and saveits right orthogonal matrix to the matrix Ve.

Step 6 Compute the small-size difference matrix, Ye=U VT.e e e

Step 7 Scale Ye by dividinga

p, where =NT/Ne.

Fig. 4. Water rate (STB/day) at the well Prod-1Step 8 Adjust Ye to ensure that its mean is zero and itsvariance meets the requirement.

Step 9 Use Ye to start the EnKF loop.

To note that when large models are considered, theSVD of Y may take substantial amount of timebecause the length of state vector becomes quite large.Thus, some special SVD schemes need to be considered,for example, parallel SVD, to keep the computationalefficiency of the Improved Initial Member Samplingalgorithm (Reviewer 1, Comment 6).

5. Synthetic case study

This is a 2D, 2-phase, water flooding problem. Thereservoir simulation model has 1616 gridblocks, eachof which has equal volume, 606040 ft3. There are 5wells: 1 injector at the center and 4 producers at the fourcorners. All 5 wells have constant bottomhole pressuresso that only variable water injection rate and wateroilproduction rates are available to be assimilated asproduction data in the EnKF. The reservoir produces200 days. The first seismic survey is at day 1 and the

second is at day 198. The production data, 5 water rates

before and after the EnKF (M128 group).

and 4 oil rates, are assimilated every 10 days from day10 to day 200. The seismic impedance data are assumedto be available at each gridblock so the total number ofseismic data for each seismic survey is 256.

Both the true permeability field and the true porosityfield that provide the observed seismic and productiondata were generated using the Cholesky decompositionof the model covariance matrix (Oliver, 1994; Ripley,1981). The variogram model used to construct the

covariance matrix is an exponential model with its twoprinciple directions along 90 and 0 directions (Deutschand Journel, 1992). The ranges are 9 and 6 gridblocksrespectively. The log permeability mean is 5.5 (245 md)and the porosity mean is 0.2. The standard deviation ofthe log permeability is 0.5 and that of the porosity is0.02. The correlation coefficient between the logpermeability and porosity is 0.5. The same variogrammodel was also used to generate the initial ensemble

88 Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399Fig. 5. Oil rates (STB/day) at the producers before and after the EnKF (M128 group).

Fig. 6. Cross plots of seismic impedance (M128 group).

89Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399members by inputting different random numberssampled from N(0, 1) (Oliver, 1994; Ripley, 1981)(Reviewer 2, Comment 7). Fig. 1 shows the true logpermeability and porosity fields. The 4 black points inFig. 1(a) stand for the 4 producers and the centered graypoint denotes the injector. The water saturationdistribution at day 200 is shown in Fig. 2. The producerat the left lower corner (hereafter well Prod-1) has waterbreakthrough.

Two ensembles were used as initial members for theEnKF independently to compare the effects fromimproved initial member sampling algorithm. Oneensemble has 128 members generated by directly

sampling the prior pdf. The other one also has 128members, which were re-generated from a 256-member

Fig. 7. Final mean of log permeabiligroup. They are called M128 group and M128F256group respectively in the following sections. In the twocases using those two groups, same observations wereused. Local data assimilation was used for both cases toassimilate the first seismic data set.

5.1. Results from M128 group

In Fig. 3, water injection rate at the injector isshown, where the red lines go through all observedwater injection rates and all blacks lines are computeddata from all ensemble members. Following thetraditions in reservoir simulation, water injection rate

is shown as negative. Before the EnKF, it can be seenthat the variation in injection rates is large, especially at

ty and porosity (M128 group).

the early time (Fig. 3(a)). After the EnKF, the injectionrates at all ensemble members become closer to the truerate, see Fig. 3(b).

Fig. 4 shows the water production rates before andafter the EnKF in the well Prod-1. From the red lines, itcan be seen that the true model has water breakthrougharound day 180. Before the EnKF, a few of ensemblemembers cannot capture the correct timing, see Fig. 4(a). After the EnKF, all ensemble members have bothcorrect water breakthrough time and water productionrate afterward (Fig. 4(b)). The three other wells do nothave water breakthrough, so only oil production ratesneed to be honored. Fig. 5 shows the oil productionrates at all 4 production wells. After the EnKF(Reviewer 1, Comment 7), the oil production ratesfrom all ensemble members are distributed more

for the large difference is that seismic impedance datahave higher sensitivity to porosity than to permeability(Dong and Oliver, 2002). This difference is moreobvious from the cross plots shown in Fig. 8. The redlines in Figs. 6 and 8 are linear regression lines. All havethe same forms, Ytrue/obs=A+BXest./comp. Values ofcoefficients A, B and correlation coefficients R are listed

Table 2Regression parameters of porosity, log permeability, and impedance(M128 group)

Parameter A B R

Porosity 0.00299 1.0146 0.97927Log permeability 2.90479 0.45336 0.42081First seismic impedance 6.333105 1.09592 0.99975Second seismic impedance 8.435104 1.01278 0.96781

90 Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399closely around the observations.The two seismic impedance data are also honored

very well after the EnKF, which can be seen from Fig. 6.The mean of seismic impedance data from all ensemblemembers after the EnKF has very strong correlation withthe observed impedance data. The second impedancecross plot has more scattered points because in thisexample, the initial saturation and pressure wereassumed to be known so that pressure and saturationdistributions have higher uncertainties at the secondseismic survey time than those at the first survey time.

The final mean of permeability and porosity fieldsafter assimilating all data are shown in Fig. 7. Comparedwith the truth shown in Fig. 1, it can be seen that theporosity field has been recovered very well. Only a fewfeatures of the true permeability field are captured in theestimate, for example, the high permeability spotbetween the injector and the well Prod-1. The reasonFig. 8. Cross plots of permeabilityin Table 2. The regression parameters also show thatseismic data have been honored very well; porosity fieldhas been estimated much better than the permeabilityfield.

The EnKF continuously adjusts ensemble membersby assimilating observation data to make the mean ofensemble members closer to the truth. Hence, thedeviation between ensemble members and the truth ateach gridblock can measure how close the estimationsare to the true models. The deviation of each variablefrom the truth is defined as

ri 1Ne

XNej1

yi;jytrue;i2 i 1; 2;: : :;Nm ;vuut 8

where Ne is the number of ensemble members and Nm isthe number of gridblocks. It is expected to get smallerand porosity (M128 group).

ScienY. Dong et al. / Journal of Petroleumwith more data assimilation. In Fig. 9, evolution ofpermeability deviation from the truth is shown. To have abetter comparison, all plots use the same color scale,where blue color stands for the lowest value and red color

Fig. 9. Evolution of permeability deviat91ce and Engineering 53 (2006) 8399for the highest value. The initial ensemble members havehigh deviation from the truth so most of the field is full ofgreen and red colors, see Fig. 9(a). At day 1, after the firstseismic data assimilation, the spot previously filled with

ion from the truth (M128 group).

red color in the left upper corner becomes smaller and thered color changes into yellow color; more gridblockshave light blue colors, which shows that seismic dataassimilation helps to adjust permeabilities of the

ensemble members toward the true values (Fig. 9(b)).At day 10, the first production data assimilation day,more gridblocks reduce their deviation and the red spotin left upper corner almost disappears (Fig. 9(c)). There

92 Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399Fig. 10. Evolution of porosity deviation from the truth (M128 group).

is reduction at the center because the water injection ratewas used. Then, as more data are assimilated, morereduction is obtained, see Fig. 9(d) and (e). In Fig. 9(f)and (g), however, some high deviation blocks appearagain, which are from over-adjustments around waterbreakthrough time at the well Prod-1. Fortunately, theyare reduced by assimilating the second seismic imped-ance data, see Fig. 9(h).

The same deviation in Eq. (8) is also computed forporosity and shown in Fig. 10. Same color scale is usedto provide a clear comparison along the time axis. Theinitial deviation map has only red color, see Fig. 10(a),which shows that even the smallest deviation value inthe initial map is larger than the biggest value in day198, see Fig. 10(h). As can be seen from permeabilitydeviation evolution, the porosity also experiences

porosity field after seismic data assimilation at day 1 is

Fig. 12. Mean porosity field after seismic data assimilation at day 1(M128 group).

93Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399reduction with more data assimilated, see Fig. 10(b) to(e). The two seismic impedance data assimilations haveobvious effects: the first one reduces the deviation in allgridblocks, which makes the red color in the initial mapdisappear completely (Fig. 10(b)), and the second onereduces the high deviation blocks that come from over-adjustments around water breakthrough time at the wellProd-1, see Fig. 10(h), (f), and (g).

Another effective tool to measure the EnKF behavioris called the integrated Root Mean Square (RMS) error,which is a distance between ensemble mean and thetruth. The definition is in Eq. (9),

RMS

1Nm

XNmi1

1Ne

XNej1

yi;jytrue;i

!2vuut : 9Different from the deviation defined in Eq. (8), the RMSerror is a scalar and sums over both ensemble membersFig. 11. RMS error of permeabilityand gridblocks. The RMS error of both permeability andporosity along with time are shown in Fig. 11. From Fig.11(b), it can be seen that after the first seismicimpedance data assimilation at day 1, the porosityRMS error drops about one order of magnitude, which isalso reflected through the dramatic reductions in thedeviation maps, see Fig. 10(a) and (b). Between day 1and day 170, however, production data assimilationdoes not provide any substantial changes to the RMSerror, which can also be seen from the deviation mapsshown in Fig. 10(c) to (e), where patterns in all maps arebarely changed. The reason is that seismic impedancedata are so sensitive to porosity that even one dataassimilation has been able to adjust the ensemble meanto be very close to the true porosity field. The meanand porosity (M128 group).

r befo

94 Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399shown in Fig. 12. Comparing it with the true porosityfield and the mean porosity field at day 200 shown inFigs. 1(b) and 7(b), it is clear that the porosity estimateat day 1 is very good. Hence, subsequent productiondata assimilation will not provide much information.Around water breakthrough time at the well Prod-1, day180 to day 190, the porosity RMS error increases to ahigh value, which again decreases after the secondseismic impedance data assimilation at day 198.

For the permeability RMS error, see Fig. 11(a), thefirst seismic data assimilation provides a small reduc-tion. It is understandable that the reduction in perme-ability error is small because at day 1, the flow field ispoorly developed and it is hard for seismic data tocapture permeability. Between day 1 and day 180,production data assimilation results in a small steadyreduction in the RMS error. That is why in Fig. 9(b) to(e), more and more gridblocks change colors from greento blue. Around water breakthrough time at the wellProd-1, very dramatic oscillations appear. After thesecond seismic data assimilation, the permeability RMSerror is reduced back to a value that is just a little bitsmaller than the initial one. Note that the RMS error isan average of all ensemble members at all gridblocks sothat a few large values can have a large effect on the

Fig. 13. Water rate (STB/day) at the injectovalue. In Fig. 9(h), about one third of gridblocks are still

Fig. 14. Water rate (STB/day) at the well Prod-1 bein green colors although other blocks are in blue, whichpartly explains why improvements in the RMS error plotare less obvious than in the deviation map.

In this M128 case, both production data and time-lapse seismic impedance data have been honored verywell. The porosity field has been recovered successfully.However, the permeability estimate is poor. The RMSerror of permeability estimate does not reduce muchafter assimilating all data and has severe oscillationaround water breakthrough time. This problem can besolved by increasing the size of the ensemble or by usingthe improved initial member sampling algorithm(Reviewer 1, Comment 9).

5.2. Results from M128F256 group

Water injection rate at the injector is shown in Fig.13, both before and after the EnKF. As before, red linesstand for observation and black lines are for ensemblemembers. Since water injection rate is also honoredquite well in M128 case, there is not much differencebetween Figs. 13(b) and 3(b).

Water production rate at the well Prod-1 before theEnKF and after the EnKF is shown in Fig. 14. After theEnKF, water production rate is honored very well, too

re and after the EnKF (M128F256 group).(Fig. 14(b)). Oil production rates at all four production

fore and after the EnKF (M128F256 group).

ic imp

Y. Dong et al. / Journal of Petroleum Scienwells before and after the EnKF have only minorimprovements compared to the M128 case, so they arenot shown here. Obviously, the production historymatching is good enough in the M128 case and benefitslittle from the improved initial member samplingalgorithm.

Seismic data matching does not gain obviouschanges either (see cross plots in Fig. 15). Thecorrelation between the observed seismic impedanceand the seismic impedance mean at the time of the firstseismic survey is very strong (Fig. 15(a)). The secondone has relatively more scattered points, but still is wellcorrelated. The small improvements are reflected

Fig. 15. Cross plots of seismthrough correlation coefficients of the regression lines(red lines in Fig. 15), listed in Table 3.

The final mean of permeability field and porosityfield are shown in Fig. 16. Similar to the M128 case,porosity mean (Fig. 16(b)) is well estimated of the truefield (Fig. 1(b)). Even most of the detailed features arerecovered by the EnKF. The permeability mean (Fig. 16(a)), however, looks much different from its counterpartshown in Fig. 7(a). The big blue spot at upper left cornerof Fig. 7(a) is replaced with a small narrow blue stripe inthe same place of Fig. 16(a). At the lower left corner in

Table 3Regression parameters of porosity, log permeability, and impedance(M128F256 group)

Parameter A B R

Porosity 0.00483 1.02319 0.97822Log permeability 1.79454 0.67076 0.57235First seismic impedance 6.035105 1.09131 0.99987Second seismic impedance 8.559104 1.01359 0.96793Fig. 16(a), more gridblocks have high permeabilityvalues, which is also an improvement.

Cross plots of permeability and porosity between thetruth and the mean are shown in Fig. 17. It can be seenthat correlation between the permeability mean and thetrue permeability is better than it was for the M128 case,i.e., points are less scattered (Fig. 17(a)), which isclearer from Table 3. All regression lines use the sameforms: Ytrue/obs=A+BXest./comp.

Using the same definition of the deviation in Eq. (8),evolution of permeability deviation from the truth isshown in Fig. 18. To have a legitimate comparison to thesame type of plots in the M128 case, Fig. 18 uses the

edance (M128F256 group).

95ce and Engineering 53 (2006) 8399same color scale. Comparing to the M128 case (Fig. 9(a)), the initial permeability deviation in M128F256 case(Fig. 18(a)) has no great improvements. There are tworeasons for the similarity: (1) improved initial samplingmethod uses the same prior pdf as conventionalsampling method; (2) the deviation does not directlymeasure distances among the ensemble members soimprovements are not obvious although the spacespanned by resampled members is larger than thatspanned by members without resampling. At day 1, thefirst seismic data are assimilated, which reducesdeviations in some blocks, see Fig. 18(b). Comparedwith Fig. 9(b), the general features of Fig. 18(b) arealmost identical because at day 1, seismic data are notsensitive directly to permeability field or indirectlythrough saturation because flow has not yet occurred.However, at locations with deviation reduction, it can beseen that the magnitude of reduction is larger in theM128F256 case than in the M128 case, i.e., light blue isreplaced with dark blue. This better adjustment comes

Fig. 16. Final mean of log permeability and porosity (M128F256 group).

96 Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399from better approximation to the covariance matrixgained from the resampling algorithm. At day 10, thefirst production data are assimilated. The reduction inthe deviation is greater in this case (Fig. 18(c)) than thatin the M128 case (Fig. 9(c)), especially at the lower leftcorner and an area to the left side of the injector. Similarfeatures can be seen from Figs. 18(d) and 9(d). The mostimportant improvements obtained from resampling theinitial members are around day 180 and day 190, seeFig. 18(f) and (g). Remember that some blocks havehigh deviations again at those 2 days in the M128 casedue to over-adjustments around water breakthroughtime at the well Prod-1 (Fig. 9(f) and (g)). In this case,however, there is no obvious increase in deviation,Fig. 17. Cross plots of permeability awhich shows that the resampled initial ensemblemembers have sufficient degrees of freedom to capturethe true model parameters even when large changes inproperties are required at water breakthrough. Aftersecond seismic data assimilation at day 198, permeabil-ity deviation has more reductions (Fig. 18(h)) comparedwith M128 case (Fig. 9(h)), i.e., more regions have darkblue color. The high sensitivity of seismic impedancewith respect to porosity makes the porosity honoredvery well in M128 case so the porosity deviation inM128F256 case does not improve much and is notshown here.

It can be seen that the resampled initial ensemblemembers have better filter behavior, especially aroundnd porosity (M128F256 group).

water breakthrough time. The over-adjustment problemcan be mitigated to some extent by using the resampledmembers. Fig. 19 shows the RMS error defined in Eq.

(9). It is clearer from Fig. 19(a) that filter behaviorusing the resampled initial ensemble members is betterthan the M128 case (Fig. 11(a)): (1) there is continuous

97Y. Dong et al. / Journal of Petroleum Science and Engineering 53 (2006) 8399Fig. 18. Evolution of permeability deviation from the truth (M128F256 group).

lity an

Scienreduction in the RMS error between the two seismicsurveys due to production data assimilation; (2)instability around water breakthrough time is notsevere any more. The plot of the RMS error of porosity(Fig. 19(b)) still has similar feature as in the M128 case(Fig. 11(b)). The first seismic data assimilation gives alarge reduction of the RMS error. The subsequentproduction data assimilation provides very smalladditional reduction to the RMS error because theestimate is already close to the truth. Around waterbreakthrough time, there are some oscillations, but thesecond seismic data assimilation returns the RMS errorto a low value comparable with the one before water

Fig. 19. RMS error of permeabi

98 Y. Dong et al. / Journal of Petroleumbreakthrough.

6. Conclusions

The small synthetic case study showed that the EnKFis a possible alternative method for automatic historymatching both production data and time-lapse seismicdata. Both production data and seismic impedance datawere honored very well. The permeability field was notwell constrained from the seismic data. The estimate ofthe porosity field from seismic data integration wasquite good because seismic impedance is very sensitiveto porosity. Production data are necessary to providesensitivities to recover the permeability field and toprovide constraints for the simulator.

128 ensemble members were enough for this smallcase. For large scale cases, however, a larger ensemblemay be required. Determination of the size requiresmore investigation. When seismic data are considered,research based on some large scale problems areessential to determine if the EnKF with seismic data isscalable.

For this small case, the overall cost is 128 simulationruns plus overhead of matrices computation. However,for large scale problems, 128 members may be verydemanding in computation resources. Hence, to reducethe number of ensemble members is a crucial issue forthe EnKF. Therefore, to carefully select initial membersby using improved sampling algorithm is highlynecessary because resampled members can usuallyprovide large covariance space at the expense of smallensemble size.d porosity (M128F256 group).

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Sequential assimilation of 4D seismic data for reservoir description using the ensemble Kalman.....IntroductionEnsemble Kalman filterIntegration of time-lapse seismic impedance dataImproved initial member samplingSynthetic case studyResults from M128 groupResults from M128F256 group

ConclusionsReferences