SPE 84372

Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter

Geir Nævdal, RF-Rogaland Research; Liv Merethe Johnsen, SPE, Norsk Hydro; Sigurd Ivar Aanonsen∗, SPE, Norsk Hydro;Erlend H. Vefring, SPE, RF-Rogaland Research

Copyright 2003, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the SPE Annual Technical Conference andExhibition held in Denver, Colorado, USA, 5 – 8 October 2003.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subjectto correction by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presentedat SPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this pa-per for commercial purposes without the written consent of the Society of Petroleum Engi-neers is prohibited. Permission to reproduce in print is restricted to an abstract of not morethan 300 words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractThe use of ensemble Kalman filter techniques for continuousupdating of reservoir model is demonstrated. The ensembleKalman filter technique is introduced, and thereafter applied ontwo 2-D reservoir models. One is a synthetic model with twoproducers and one injector. The other model is a simplified 2-Dfield model, which is generated by using a horizontal layer froma North Sea field model.

By assimilating measured production data, the reservoirmodel is continuously updated. The updated models give im-proved forecasts. Both dynamic variables, as pressure and sat-urations, and static variables as the permeability are updated inthe reservoir model.

IntroductionIn the management of reservoirs it is important to utilize allavailable data in order to make accurate forecasts. For shorttime forecasts, in particular, it is important that the initial val-ues are consistent with recent measurements. The ensembleKalman filter1 is a Monte Carlo approach, which is promisingwith respect to achieving this goal through continuous modelupdating and reservoir monitoring.

In this paper, the ensemble Kalman filter is utilized to up-date both static parameters, such as the permeability, and dy-namic variables, such as the pressure and saturation of the reser-voir model. The filter computations are based on an ensembleof realizations of the reservoir model, and when new measure-ments are available new updates are obtained by combining themodel predictions with the new measurements. Statistics aboutthe model uncertainty is built from the ensemble. While newmeasurements become available, the filter is used to update all

the realizations of the reservoir model. This means that an en-semble of updated realizations of the reservoir model is alwaysavailable.

The ensemble Kalman filter has previously been success-fully applied for large-scale nonlinear models in oceanography2

and hydrology3. In those applications only dynamic variableswere tuned. Tuning of model parameters and dynamic variableswas done simultaneously in a well flow model used for under-balanced drilling4. In two previous papers5,6, the filter has beenused to update static parameters in near-well reservoir models,by tuning the permeability field. In this paper, the filter hasbeen further developed to tune the permeability for simplifiedreal field reservoir simulation models.

We present results from a synthetic model as well as a sim-plified real field model. The measurements are well bottom-hole pressures, water cuts and gas/oil ratios. A synthetic modelgives the possibility of comparing the solution obtained by thefilter to the true solution, and the performance of the filter canbe evaluated. It is shown how the reservoir model is updatedas new measurements becomes available, and that good fore-casts are obtained. The convergence of the reservoir propertiesto the true solution as more measurements becomes available isinvestigated.

Since the members of the ensemble are updated indepen-dently of each other, the method is very suitable for parallelprocessing. It is also conceptually straightforward to extend themethodology to update other reservoir properties than the per-meability.

Based on the updated ensemble of models, production fore-casts and reservoir management studies may be performed ona single ”average” model, which is always consistent with thelatest measurements. Alternatively, the entire ensemble may beapplied to estimate the uncertainties in the forecasts.

Updating reservoir models with ensemble Kalman fil-terThe Kalman filter was originally developed to update the statesof linear systems to take into account available measurements7.In our case, the system is a reservoir model, using black oil, andthree phases (water, oil and gas). For this model, the solutionvariables of the system are the pressure, the water saturation, inaddition to a third solution variable that depends on the oil andgas saturation. If the gas saturation is zero, the third solutionvariable becomes the solution gas/oil ratio, if the oil saturation

∗Now with Centre for Integrated Petroleum Research, University of Bergen

2 SPE 84372

is zero it becomes the vapor oil/gas ratio. Otherwise the thirdsolution variable is the gas saturation. The states of this systemare the values of the solution variables for each grid block of thesimulation model. This model is non-linear.

An early attempt to extend the ideas of the Kalman filter tonon-linear models is the extended Kalman filter which is basedon linearization of the non-linear model. The extended Kalmanfilter is however not suitable for very large models and also failsif the non-linearities are too severe. Therefore an alternative isneeded1. The ensemble Kalman filter has shown to be a promis-ing approach for large scale non-linear atmospheric and oceano-graphic models.

For the reservoir model, however, there are poorly deter-mined model parameters that have a governing influence. Inthis study we will focus on using the ensemble Kalman filterto update the permeability, in addition to the above mentionedstate variables. This means that we extend the state vector withthe permeability of each grid block (assuming that the perme-ability is isotropic). Note that in principle any input parameterin the model can be updated.

The literature on estimating dynamic variables and modelparameters simultaneously for non-linear models using somevariants of a Kalman filter is modest. The topic is discussedby Wan and Nelson8. They give further references, but do notdiscuss the use of ensemble Kalman filter to solve such a task.

The ensemble Kalman filter has previously been used to up-date both dynamic variables and model parameters for a two-phase well flow model used in underbalanced drilling4,9. Theapproach has also been used for updating the permeability anddynamic variables in a two-phase near-well reservoir setting5,6.

The ensemble Kalman filter is based on a Monte-Carlo ap-proach, using an ensemble of model representations to evalu-ate the necessary statistics. We have used 100 members in theensemble, based on experience from atmospheric data assimila-tion, where this have been sufficient10.

The filter consists of sequentially running a forecast stepfollowed by an analysis step. The input to the forecast step isthe result obtained from the analysis step, which is an updateddescription of the model after assimilating a new set of mea-surements. The forecast step consists of running the reservoirsimulator for each of the model realizations. The forward sim-ulations are ending at the next point in time where new mea-surements are to be assimilated. The state vector after runningthe forecast step is denoted by s

fk and the state vector after the

analysis step is denoted by sak.

The filter is initialized by generating an initial ensemble.The generation of the initial ensemble can be described by theequation

sa0,i = sa

0+ em

0,i, (1)

where i runs from 1 to the number of members in the ensemble.Here sa

0is the mean of the initial ensemble. The terms em

0,i aredrawn from a mixture distribution with zero mean (we use theterm mixture distribution11 if a random variable have a distribu-tion that depends on a quantity which also have a distribution).Further details on the distribution that we use in generating theterms em

0,i is presented below.The forecast step consists of running the model (i.e. the

simulator solving the reservoir model) and giving an expressionfor the uncertainty in the model output. In these studies we

have used a commercial reservoir simulator. We denote runningthe reservoir simulator forward to the next point in time wheremeasured data is going to be assimilated by f . The reservoirsimulator is run once for each member of the ensemble. Forthe i’th member of the ensemble at time level k, we denote theforecasted state vector by s

fk,i and the analyzed state by sa

k,i. Inthe forecast step the simulator is run from the current time (saytime level k−1) to the point in time where the next measurementbecomes available (time level k). To take into account the modeluncertainty, we add model noise to the ensemble members priorto each simulation. This gives the equation

sfk,i = f(sa

k−1,i + emk,i). (2)

The terms emk,i is drawn from a mixture distribution.

Generation of the initial ensemble and the generation of themodel noise at each step is done along the same lines, repre-sented by the terms em

0,i and emk,i in Equations 1 and 2, respec-

tively. The model noise is basically placed in the permeability.The noise added to the permeability is independent of the noiseadded to the time dependent variables of the model (pressure,saturations, etc.).

To generate stochastic realizations of the permeability fieldwe assume that the permeability is spatially correlated with aGaussian correlation model, i.e. that the permeability in gridblock (i1, j1) is correlated with the permeability in grid block(i2, j2) with correlation coefficient

e−( i1−i2l )

2

−( j1−j2l )

2

. (3)

Using the correlations specified in Equation 3 a correlation ma-trix C is computed. From this correlation matrix, the covariancematrices used while generating the initial permeability field andthe model noise is computed as σ2C, where σ is the standarddeviation in the permeability of each grid block. The standarddeviation used while generating the initial permeability field(Eq. 1) is usually much higher than the used while adding themodel noise in Eq. 2. The correlation length, l, is also treatedas a normally distributed stochastic variable. This means that(the noise added to) each ensemble member is generated by firstdrawing l from a normal distribution, then generating C and fi-nally draw a permeability field from a multinormal distributionwith zero mean and covariance matrix σ2C.

To generate the noise added to time dependent variablesof the model, we draw from a normal distribution with zeromean and covariance matrix Qd. In the examples we have usedQd = εI, where ε ≈ 2 · 10−16. In generation of the initial en-semble there is no noise in the state variables, as all the ensem-ble members are started using the same equilibrium conditionof the reservoir.

The analyzed state at time level k is computed by takinginto account the measurement vector at time level k. The theoryassumes that there is a linear relationship between the measure-ments, dk, and the states, sk, expressed by the equation

dk = Hsk. (4)

To take into account the fact that there is a non-linear rela-tionship between the observed quantities (measurements in thewell), and the state variables, we extend the state vector by one

SPE 84372 3

state for each non-linear measurement. This brings the corre-spondence between all measurements and the state variableson the form (4). We assume that the measurement noise hasa multinormal distribution with zero mean and covariance ma-trix Rk. The index k is included since the covariance matrix ofthe measurement noise may be time dependent.

It is shown that if the measurement is not treated as a ran-dom variable, the updated ensemble will have correct mean, butto low variance12. This means that to get consistent error prop-agation in the ensemble Kalman filter one has to treat the ob-servations as random variables. This is done by using the actualmeasurement as reference, and adding random noise, reflectingour assumptions on the measurement noise. This means thatthe actual measurement dk serves as the reference observation.For each member of the ensemble an observation vector dk,i isgenerated randomly as

dk,i = dk + eok,i (5)

where eok,i is drawn from a multinormal distribution with zero

mean and covariance Rk.The error covariance matrix for the state variables of the

model is defined in terms of the true state as the expectation

E((sf

k − stk)(sf

k − stk)T

). (6)

Since the true state is not known, we approximate the true stateby the mean of the ensemble

stk ≈ s

fk =

1

N

N∑

i=1

sfk,i (7)

where N is the sample size of the ensemble. With this approxi-mation of the true state, an approximation of a left factor of theerror covariance matrix of the model is

Lfk =

1√

N − 1

[(sf

k,1 − sfk) . . . (sf

k,N − sfk)

], (8)

a matrix with N columns. The approximation of the model errorcovariance matrix then becomes

Pfk = L

fk(Lf

k)T . (9)

The expression of the Kalman gain matrix is7

Kk = PfkH

T (HPfkH

T + Rk)−1. (10)

The analyzed state of each member of the ensemble is computedas

sak,i = s

fk,i + Kk(dk,i −Hs

fk,i). (11)

The analyzed error covariance matrix, Pak, of the model can

be computed along the same lines as Pfk . Since the updating of

the ensemble is linear, the new estimate of the true state, basedon the ensemble after the analysis step is

stk ≈ sa

k = sfk + Kk(dk −Hs

fk), (12)

and the model error covariance matrix after the analysis step is

Pak = (I −KkH)Pf

k . (13)

The underlying assumptions behind the filter are that there iszero covariance between the model error and the measurementerror and that both the model error and measurement error areuncorrelated in time.

In evaluating the quality of the solution we use the rootmean square (RMS) error defined as

√√√√ 1

M

M∑

j=1

(Xj − Xt

j

)2, (14)

where M is the number of state variables, Xj is the estimate ofstate variable j, and X t

j is the true value of state variable j.

Example 1: Synthetic caseA two-dimensional 50× 50× 1 grid is defined as shown in Fig-ure 1, modeling a reservoir with a constant dip. The model con-tains two production wells, P2 and P4, and a single water injec-tor, WI. The water injector is located below the initial water-oilcontact, as shown in Figure 1.

A ”true” permeability distribution is created, where the per-meability varies linearly as a function of the x-coordinate in theinterval [200, 2000] mD. A constant value of 0.15 is used forthe porosity in all grid cells. As relative permeabilities and PVTproperties we have used values similar to those used in a NorthSea field.

The production data is generated by running the reservoirsimulator using group control for the two production wells, witha constant total liquid rate of 15.000 Sm3/d. This total rate isdistributed between the two production wells according to theirproduction potential. Voidage replacement is used for the injec-tion well, leading to a water injection rate which may vary withtime if pressure fells below the bubble point and free gas is pro-duced in either of the wells. No pressure constraints are usedfor the wells. The measurements are generated by running thesimulator with the “true” permeability, and adding noise to theresulting well pressures, oil rates, GORs and water cuts. Mea-surement are assimilated after 4 days, then after 22 days, thenonce every month for 50 months.

The measurement uncertainties are presented in Table 1. Weapply the filter to take into account the information gained from8 measurements at each assimilation, 4 for each of the produc-tion wells. The measured quantities are the pressure, the oilproduction rate, the water cut and gas/oil ratio. There are nomeasurements from the injector.

While running the ensemble Kalman filter we keep all quan-tities constant except for the permeability (which is isotropic)and the state variables, which are the pressure, the saturation ofgas and water, and the solution gas/oil ratio. In this case thereare no vaporized oil. The reservoir simulator is run in “history-matching mode”, that is, the wells are controlled by reservoirvolume rates equal to the once used when generating the (syn-thetic) history data. Thus the reservoir volume offtake is thesame for all ensemble members. The same applies for the in-jection rates. In the computation with the Kalman filter we usethe natural logarithm of the permeability instead of the perme-ability.

The ensemble consists of 100 members. The initial en-semble is generated using a mean correlation length of 16 grid

4 SPE 84372

blocks with a standard deviation of 6. The initial ln-perme-ability fields for the Kalman filter is generated using a standarddeviation of 0.5. For the model noise (Eq. 2) the standard devi-ation is set to 5 · 10−4.

For comparison we have run a reference solution using themean permeability of the initial ensemble (i.e., approximately1000mD). The initial ensemble is generated using the mean ofthe ln-permeability of the three grid blocks perforated by a well.

In Figure 2 we present the outcome of the filter for each ofthe 8 measured variables, together with the reference solution.As we can see, the filter essentially tracks all the measurements,with a small exception for the gas/oil ratio for the well P4.

In Figure 3 we show forecasts based on the ensemble meanafter 4 days, 604 days and 908 days. As expected, the quality ofthe forecasts are generally improved as more data is assimilated.It should be noted, however, that because of the pressure mea-surements, the permeability trend is recovered by the filter verysoon. Already after 4 days, the permeability in the region fromP2 to P4 is almost correct, as can be seen in Figure 4 showingthe development in the estimated permeability. After 300 days,long before any water breakthrough, the trend is correct in theentire reservoir. At later times, however, the permeability esti-mates move away from the correct values, and the effect of thelater measurements, including the water cut development, is notquite clear. Still, the uncertainty estimates shown in Figure 5seems reasonable, being lowest close to the producers, wherethe measurements are obtained, and decreasing with time. Thehighest uncertainty is close to the boundary of the reservoir.

In Figure 6 we show how the root mean square (RMS) er-ror defined in Eq. 14 evolves. Again, we see that the error inestimated permeability decreases very rapidly, before it startsincreasing. At late times the RMS error is the same as for thereference solution. Here, the correct permeability trend is prob-ably more important for the predictions than the global RMSerror, and since the trend is recovered, the forecasts will stillimprove with time. However, an increase in the RMS error, de-spite of more available data is not satisfactory, and the reasonfor this behavior should be further investigated. For the dy-namic variables, we should bear in mind that all the membersof the ensemble are initiated from a model with a known equi-librium condition, giving zero RMS error. The RMS thereforehas to increase initially. Theoretically, the error should stabilizeat a level given by the data error and model error. It is seen thatRMS error of the dynamic variables stabilize after some time,but because of the complicated relation between data, modeland state variables, it is difficult to verify that the stabilized val-ues of the RMS are correct.

Example 2: Simplified field modelA simplified 2D field model has been constructed by taking outa horizontal layer from a model of a North Sea field. The di-mensions of the simplified model are 39 × 55, but with someinactive blocks. There are 1931 active grid blocks. The simula-tion spans a time range over 3955 days. The reservoir model isshown in Figure 7.

Measurements are assimilated at least once a month, butalso when new wells are starting production, or when wells areshut in. Altogether there are 171 points in time when measure-ments are assimilated. The measurements are assimilated from

14 producing wells. Two of them are starting from the first day.The reservoir is produced by gas injection, and the producersare located along the oil-water contact. There are 4 gas injec-tors. There are no measurements from the injectors.

In addition to the 14 producers and 4 injectors there aresome dummy wells included in the simulation to adjust for flowthrough the boundary. The rates of the dummy wells are keptconstant while generating the data and running the ensembleKalman filter.

The quantities that are tuned with the Kalman filter are thehorizontal permeability, which is assumed to be the same in thex- and y-direction, together with the dynamic variables, whichare the pressure, saturation of water and gas, the solution gas/oilratio and the vapor oil/gas ratio. This means that we both havevaporized oil and dissolved gas.

The “true” model is obtained from a stochastic realizationof the field. Relative permeabilities and PVT properties weretaken from the original field model. The measurements are gen-erated by running the simulator with the “true” permeability,and adding noise to the values obtained.

While running the ensemble Kalman filter we keep all quan-tities except the state variables described above constant. Likein Example 1, the reservoir simulator is run in “history-match-ing mode”, and the natural logarithm of the permeability is usedin the computations with the Kalman filter.

The ensemble consists of 100 members. The initial en-semble is generated using a mean correlation length of 16 gridblocks with a standard deviation of 6 grid blocks. The initialln-permeability fields for the Kalman filter is generated usinga standard deviation of 0.5. For the model noise (Eq. 2) thestandard deviation is set to 5 ·10−4. The initial mean of the per-meability of the ensemble is set to 1000 for all the grid blocks.

The measurement uncertainties are as in Example 1 (Ta-ble 1). We apply the filter to take into account the informationgained from 4 measurements for each of the production wells.The measured quantities are pressure, oil production rate, watercut and gas/oil ratio.

In Figure 8 the development in the estimated permeability isshown together with the “true” permeability and the referencesolution. The reference solution is the mean of the initial en-semble, and since the initial ensemble is based on a model withconstant mean permeability of 1000 mD, the reference modelwill have a permeability of approximately 1000 mD in all cells.One can observe that after 1611 days a good match of the per-meability field is obtained, especially close to the producingwells. The location of the producers are shown with circles. Forthe permeabilities along the boundaries, the permeability startsdrifting towards the end of the simulation. Further developmentof the filter is needed to avoid this effect.

The uncertainty in the ln-permeability is presented in Fig-ure 9. The uncertainty is lowest close to the producers, wherethe measurements are obtained. The highest uncertainty is closeto the boundary of the reservoir. The uncertainty is decreasingas more measurements are assimilated.

In Figure 10 we present the outcome of the filter for some ofthe measured variables, together with the reference solution. Wehave selected to present results from measurements that haveamong the largest relative deviations between the truth and thesolution obtained from the Kalman filter.

SPE 84372 5

In Figure 11 we show forecasts based on the ensemble meanafter 4 days, 819 days and 3050 days for the same measure-ments as presented in Figure 10. As expected, the quality of theforecasts is improved as more data is assimilated.

In Figure 12 we show how the root mean square (RMS) er-ror defined in Eq. 14 evolves. Like in Example 1, the RMSerror for the permeability decreases very rapidly and then in-creases. Still the predictions clearly improve with time as addi-tional measurements become available. Note also that with theexception for RS and RV, the RMS values for the state variablesstabilize relatively fast at quite low levels.

DiscussionThe ensemble Kalman filter technology seems promising fortuning permeability and dynamic variables, such as the pressureand saturations, in reservoir simulator models. In both the ex-amples presented, the filter were able to recover the main trendsof the permeability field rapidly due to the continuous down-hole pressure measurements. However, although new measure-ments are added continuously, the error in the estimated perme-ability increases late in time. The reason for this instability isnot fully understood.

At this stage, all the variables that are not tuned, are keptfixed while running the reservoir simulator. This includes theinitial state of the reservoir. Therefore the difference betweenthe estimate obtained from the filter and the true value of thedynamic variables have to increase in the beginning. After awhile the differences stabilize.

It is crucial for a successful application of the methodologythat the ensemble of models correctly reflects the uncertainty atall times. So the generation of the initial ensemble and how toadd model noise are important questions which also have to befurther addressed.

Technically, there is no limit with respect to how manymodel parameters which can be updated, but the effect of addinga large number of parameters on filter stability, etc., has not yetbeen considered.

ConclusionsAn ensemble Kalman filter technology is developed for contin-uous updating of reservoir simulation models. Examples areshown in 2D, but the methodology can easily be extended to 3Dand applied to any existing reservoir simulator.

The filter is used to tune the permeability and dynamic vari-ables like pressure and saturations. The continuous updatingensures that the predictions always start from a solution thatmatches the observed production data.

The methodology is applied on two 2D reservoir models:One synthetic model with one injector and two producers, andone simplified field model with 14 producers and 3 injectors.In both examples the ensemble Kalman filter is able to trackthe measurements and tune the permeability field, and as moremeasurements are assimilated, the forecasts are improved.

Although the presented results are promising, this technol-ogy is novel to reservoir simulation, and further development ofthe filter is needed for this application.

AcknowledgmentsThis work has been supported financially by ENI-Agip SpA,Norsk Hydro ASA, Statoil and The Research Council of Nor-way.

NomenclatureC = correlation matrixd = measurement vectorE = expectatione = stochastic noisef = output from reservoir simulatorH = measurement matrixi = index (members of ensemble / grid block coordinates)I = identity matrixj = index (state variables / grid block coordinates)k = index (assimilation time)K = Kalman gain matrixl = correlation lengthL = left factor of covariance matrix of model unceratintyM= number of statesN = number ensemble membersP = covariance matrix of model unceratintyQ = covariance matrix of modelling errorR = covariance matrix of measurement errors = state vectorσ = standard deviationσ2= variance

Subscripts

d = dynamic variablesi = index (ensemble members)j = index (state variables)k = index (assimilating time)

Superscripts

a = analyzed (aposteriori)f = forecasted (apriori)m = model noiseT = matrix transposet = true

References1. Geir Evensen. Sequential data assimilation with a nonlinear

quasi-geostrophic model using Monte Carlo methods to forecasterror statistics. J. Geophys. Res., 99(C5):10,143–10,162, 1994.

2. Vibeke E. J. Haugen and Geir Evensen. Assimilation of SLAand SST data into OGCM for the Indian Ocean. Ocean Dynam-ics, 52:133–151, 2002.

3. Steven A. Margulis, Dennis McLaughlin, Dara Entekhabi, andSusan Dunne. Land data assimilation and estimation of soilmoisture using measurements from the Southern Great Plains1997 Field Experiment. Water Resources Research, 38(12),2002.

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4. Rolf J. Lorentzen, Kjell Kare Fjelde, Johnny Frøyen, Anto-nio C. V. M. Lage, Geir Nævdal, and Erlend H. Vefring. Under-balanced and low-head drilling operations: Real time interpre-tation of measured data and operational support. In 2001 SPEAnnual Technical Conference and Exhibition, 30 September - 3October 2001. SPE 71384.

5. Geir Nævdal, Trond Mannseth, and Erlend H. Vefring. Near-well reservoir monitoring through ensemble Kalman filter. InSPE/DOE Improved Oil Recovery Symposium, Tulsa, Okla-homa, April 2002. SPE 75235.

6. Geir Nævdal, Trond Mannseth, and Erlend H. Vefring. Instru-mented wells and near-well reservoir monitoring through en-semble Kalman filter. In Proceedings of 8th European Confer-ence on the Mathematics of Oil Recovery, Freiberg, Germany, 3- 6 September 2002.

7. Peter S. Maybeck. Stochastic Models, Estimation, and Control,volume 1. Academic Press, New York, 1979.

8. Eric A. Wan and Alex T. Nelson. Dual extended Kalman filtermethods. In Simon Haykin, editor, Kalman Filtering and NeuralNetworks, chapter 5, pages 123–173. John Wiley & Sons, Inc.,New York, 2001.

9. R. J. Lorentzen, G. Nævdal, and A. C. V. M. Lage. Tuning ofparameters in two-phase flow model using an ensemble Kalmanfilter. To appear in Int. J. Multiphase Flow.

10. P. L. Houtekamer and Herschel L. Mitchell. Data assimilationusing an ensemble Kalman filter technique. Monthly WeatherReview, 126:796–811, 1998.

11. George Casella and Roger L. Berger. Statistical Inference.Duxbury Press, Belmont, California, 1990.

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Type Std. dev.Pressure 1 barOil production 5 %GOR 5 %Water cut 5 %

Table 1– Measurement uncertainties used in both the examples.Figure 1– Example 1: Test model with true permeability field andinitial oil-water contact.

SPE 84372 7

200

250

300P2 −WBHP

BAR

SA

150

200

250P4 −WBHP

BAR

SA

0

5

10P2 −WOPR

1000

SM3

/DAY

0

5

10

15P4 −WOPR

1000

SM3

/DAY

0

0.5

1P2 −WWCT

0 500 1000 15000

0.5

1P4 −WWCT

DAYS

0 500 1000 15000

50

100

150

200P2 −WGOR

SM3/S

M3

DAYS 0 500 1000 1500

0

50

100

150

200P4 −WGOR

SM3/S

M3

DAYS

Figure 2– Example 1: Measured values (blue dots), filter solution (blue line), true solution (red stapled) and the reference value (green line) forthe measured quantities. The reference value is the solution obtained from the simulation with the mean of the initial ensemble.

200

250

300P2 −WBHP

BAR

SA

150

200

250P4 −WBHP

BAR

SA

0

5

10P2 −WOPR

1000

SM3

/DAY

0

5

10

15P4 −WOPR

1000

SM3

/DAY

0

0.5

1P2 −WWCT

0 1000 20000

0.5

1P4 −WWCT

DAYS

0 1000 20000

50

100

150

200P2 −WGOR

SM3

/SM3

DAYS 0 1000 2000

0

50

100

150

200P4 −WGOR

SM3

/SM3

DAYS

Figure 3– Example 1: Forecasted values. The forecast based on the mean of the initial ensemble (green), the forecast after 4 days (blue), theforecast after 604 days (magneta) and the forecast after 908 days (black) and the true solution (red stapled).

8 SPE 84372

Ref. sol.

200

500

1000

2000

4000

TRUE 4 days 300 days

604 days 908 days 1212 days 1518 days

Figure 4– Example 1: True and estimated permeabilities. The wells are located by small circles. The injector, WI, in the bottom, the producer P2up to the left and the producer P4 up to the right.The upper left plot shows the reference permeability which is the mean of the initial ensemble.

4 days

0

0.1

0.2

0.3

0.4

0.5

300 days 604 days

908 days 1212 days 1518 days

Figure 5– Example 1: Uncertainty in estimated ln-permeabilities

SPE 84372 9

0.4

0.5

0.6

0.7

0.8PERMX

0

2

4

6PRESSURE

0 500 1000 15000

0.05

0.1

0.15

0.2SWAT

0 500 1000 15000

0.01

0.02

0.03

0.04SGAS

0 500 1000 15000

2

4

6RS

Figure 6– Example 1: RMS error between estimated and true variables (blue) and between the reference solution and the true variables (green).PERMX is the ln-permeability.

Y Axis

Z Axis

X Axis

2195.6 2422.7 2649.8 2876.8 3103.9

Depth (m)

Figure 7– Example 2: Description of reservoir.

10 SPE 84372

Ref. sol.

200

500

1000

2000

4000

8000

TRUE 4 days 819 days

1611 days 2361 days 3050 days 3955 days

Figure 8– Example 2: True and estimated permeabilities. The reference solution is the mean of the initial ensemble (before assimilating anymeasurements). The location of the producers where the measurements are obtained are shown by circles. In the plot of the estimatedpermeabilities, only the wells that have been producing prior to the time of obtaining the estimate are included.

4 days

0

0.1

0.2

0.3

0.4

0.5

819 days

0

0.1

0.2

0.3

0.41611 days

0

0.05

0.1

0.15

0.2

2361 days 3050 days 3955 days

Figure 9– Example 2: Uncertainty in estimated ln-permeabilities. The plots in the bottom row use the same color as the upper right plot.

SPE 84372 11

0

500

1000

1500W10A −WOPR

SM3/D

AY

0

500

1000

1500W11A −WOPR

0

200

400

600W18A −WOPR

0

1000

2000

3000

4000W05C −WGOR

SM3/S

M3

0

500

1000

1500W06A −WGOR

0

1000

2000

3000W10A −WGOR

0 2000 40000

1000

2000

3000

4000W11A −WGOR

SM3/S

M3

DAYS 0 2000 4000

0

5000

10000W18A −WGOR

DAYS 0 2000 4000

0

2000

4000

6000W36A −WGOR

DAYS

Figure 10– Example 2: Measured values (blue dots), filter solution (blue line), true solution (red stapled) and the reference solution (green) forsome measured quantities. The true solution is mostly hidden behind the filter solution.

0

500

1000

1500

2000W10A −WOPR

SM3

/DAY

0

500

1000

1500W11A −WOPR

0

200

400

600W18A −WOPR

0

1000

2000

3000

4000W05C −WGOR

SM3

/SM3

0

500

1000

1500W06A −WGOR

0

1000

2000

3000W10A −WGOR

0 2000 40000

1000

2000

3000

4000W11A −WGOR

SM3

/SM3

DAYS 0 2000 4000

0

5000

10000W18A −WGOR

DAYS 0 2000 4000

0

2000

4000

6000W36A −WGOR

DAYS

Figure 11– Example 2: Forecasted values for some of the measured quantities. The forecast based on the mean of the initial ensemble (green),the forecast after 4 days (blue), after 819 days (magneta), after 3050 days (black) and the true solution (red stapled).

12 SPE 84372

0.2

0.4

0.6

0.8

1PERMX

0

2

4

6PRESSURE

0

0.02

0.04

0.06SWAT

0 2000 40000

0.02

0.04

0.06SGAS

0 2000 40000

5

10RS

0 2000 40000

0.5

1

1.5x 10−5 RV

Figure 12– Example 2: RMS error between estimated and true variables (blue) and between the reference solution and the true variables (green).PERMX is the ln-permeability.