SPE-84372-PA

Reservoir Monitoring and ContinuousModel Updating Using Ensemble

Kalman FilterGeir Nævdal, SPE, RF-Rogaland Research; Liv Merethe Johnsen, SPE, and Sigurd Ivar Aanonsen,* SPE,

Norsk Hydro; and Erlend H. Vefring, SPE, RF-Rogaland Research

SummaryThe use of ensemble Kalman filter techniques for continuous up-dating of a reservoir model is demonstrated. The ensemble Kalmanfilter technique is introduced, and thereafter applied to a simplified2D field model, which is generated by using a single horizontallayer from a North Sea field model. By assimilating measuredproduction data, the reservoir model is continuously updated. Theupdated models give improved forecasts and the forecasts improveas more data are included. Both dynamic variables, such as pres-sure and saturations, and static variables, such as permeability, areupdated in the reservoir model.

IntroductionIn the management of reservoirs, it is important to use all availabledata in order to make accurate forecasts. For short-time forecasts,in particular, it is important that the initial values are consistentwith recent measurements. The ensemble Kalman filter1 is aMonte Carlo approach, and is promising with respect to achiev-ing this goal through continuous model updating and reser-voir monitoring.

In this paper, the ensemble Kalman filter is used to update bothstatic parameters, such as permeability, and dynamic variables,such as pressure and saturations, of the reservoir model. The filtercomputations are based on an ensemble of realizations of the res-ervoir model, and when new measurements are available, newupdates are obtained by combining the model predictions with thenew measurements. Statistics about the model uncertainty are builtfrom the ensemble. When new measurements become available,the filter is used to update all the realizations of the reservoirmodel. This means that an ensemble of updated realizations of thereservoir model is always available.

The ensemble Kalman filter has previously been successfullyapplied for large-scale nonlinear models in oceanography2 andhydrology.3 In those applications, only dynamic variables weretuned. Tuning of model parameters and dynamic variables wasdone simultaneously in a well flow model used for underbalanceddrilling.4 In two previous papers,5,6 the filter has been used toupdate static parameters in near-well reservoir models by tuningthe permeability field. In this paper, the filter has been furtherdeveloped to tune the permeability for simplified real-field reser-voir simulation models.

We present results from a synthetic, simplified real-fieldmodel. The measurements are well bottomhole pressures, watercuts, and gas/oil ratios. A synthetic model gives the possibility ofcomparing the solution obtained by the filter to the true solution,and the performance of the filter can be evaluated. It is shown howthe reservoir model is updated as new measurements becomeavailable, and that good forecasts are obtained. The convergenceof the reservoir properties to the true solution as more measure-ments become available is investigated.

Because the members of the ensemble are updated indepen-dently of each other, the method is very suitable for parallel pro-cessing. It is also conceptually straightforward to extend the meth-odology to update reservoir properties other than permeability.

Based on the updated ensemble of models, production forecastsand reservoir-management studies may be performed on a single“average” model, which is always consistent with the latest mea-surements. Alternatively, the entire ensemble may be applied toestimate the uncertainties in the forecasts.

Updating Reservoir Models With EnsembleKalman FilterThe Kalman filter was originally developed to update the states oflinear systems to take into account available measurements.7 In ourcase, the system is a reservoir model, using black oil and threephases (water, oil, and gas). For this model, the solution variablesof the system are pressure and water saturation, in addition to athird solution variable that depends on the oil and gas saturation.If the gas saturation is zero, the third solution variable becomes thesolution gas/oil ratio; if the oil saturation is zero, it becomes thevapor oil/gas ratio. Otherwise, the third solution variable is the gassaturation. The states of this system are the values of the solutionvariables for each gridblock of the simulation model. This modelis nonlinear.

An early attempt to extend the ideas of the Kalman filter tononlinear models is the extended Kalman filter, which is based onlinearization of the nonlinear model. The extended Kalman filteris, however, not suitable for very large models and also fails if thenonlinearities are too severe. Therefore, an alternative is needed.1

The ensemble Kalman filter has shown to be a promising approachfor large-scale nonlinear atmospheric and oceanographic models.

For the reservoir model, however, there are poorly determinedmodel parameters that have a governing influence. In this study,we will focus on using the ensemble Kalman filter to update thepermeability, in addition to the previously mentioned state vari-ables. This means that we extend the state vector with the perme-ability of each gridblock (assuming that the permeability is isotro-pic). Note that in principle, any input parameter in the model canbe updated.

The literature on estimating dynamic variables and model pa-rameters simultaneously for nonlinear models using some variantsof a Kalman filter is modest. The topic is discussed by Wan andNelson.8 They give further references, but do not discuss the use ofensemble Kalman filter to perform such a task.

The ensemble Kalman filter has previously been used to updateboth dynamic variables and model parameters for a two-phase wellflow model used in underbalanced drilling.4,9 The approach hasalso been used for updating the permeability and dynamic vari-ables in a two-phase near-well reservoir setting.5,6

The ensemble Kalman filter is based on a Monte Carlo ap-proach, using an ensemble of model representations to evaluate thenecessary statistics. We have used 100 members in the ensemble,based on experience from atmospheric data assimilation, wherethese have been sufficient.10

The filter consists of sequentially running a forecast step fol-lowed by an analysis step. The input to the forecast step is theresult obtained from the analysis step, which is an updated de-scription of the model after assimilating a new set of measure-

* Now with Centre for Integrated Petroleum Research, U. of Bergen

Copyright © 2005 Society of Petroleum Engineers

This paper (SPE 84372) was first presented at the 2003 SPE Annual Technical Conferenceand Exhibition, Denver, 5–8 October, and revised for publication. Original manuscript re-ceived for review 28 January 2004. Revised manuscript received 8 November 2004. Paperpeer approved 18 January 2005.

66 March 2005 SPE Journal

ments. The forecast step consists of running the reservoir simulatorfor each of the model realizations. The forward simulations end atthe next point in time where new measurements are to be assimi-lated. The state vector after running the forecast step is denoted bys�k

f , and the state vector after the analysis step is denoted by s�ka.

The filter is initialized by generating an initial ensemble. Thegeneration of the initial ensemble can be described by the equation

s�o,ia = s�0

a + e�0,im , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where i runs from 1 to the number of members in the ensemble.Here, s�o

a is the mean of the initial ensemble. The terms e�m0,i are

drawn from a mixture distribution with zero mean (we use the term“mixture distribution”11 if a random variable has a distribution thatdepends on a quantity that also has a distribution). Further detailson the distribution that we use to generate the terms e�m

0,i are pre-sented here.

The forecast step consists of running the model (i.e., the simu-lator solving the reservoir model) and giving an expression for theuncertainty in the model output. In these studies, we have used acommercial reservoir simulator. We denote running the reservoirsimulator forward to the next point in time where measured dataare going to be assimilated by f. The reservoir simulator is run oncefor each member of the ensemble. For the ith member of theensemble at time level k, we denote the forecasted state vector bys� f

k,i and the analyzed state by s�ak,i. In the forecast step, the simulator

is run from the current time (say, time level k–1) to the point intime where the next measurement becomes available (time level k).To take into account the model uncertainty, we add model noise tothe ensemble members before each simulation. This givesthe equation

s�k,if = f�s�k−1,i

a + e�k,im �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

The terms e�mk,i are drawn from a mixture distribution.

The generation of the initial ensemble and the generation of themodel noise at each step is done along the same lines, representedby the terms e�m

0,i and e�mk,i in Eqs. 1 and 2, respectively. The model

noise is basically placed in the permeability. The noise added tothe permeability is independent of the noise added to the time-dependent variables of the model (e.g., pressure and saturations).

To generate stochastic realizations of the permeability field, weassume that the permeability is spatially correlated with a Gaussiancorrelation model—for example, that the permeability in gridblock(i1, j1) is correlated with the permeability in gridblock (i2,j2) withcorrelation coefficient

exp�−�i1 − i2

l �2

− �j1 − j2

l �2�. . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Using the correlations specified in Eq. 3, a correlation matrix [C]is computed. From this correlation matrix, the covariance matricesused while generating the initial permeability field and the modelnoise are computed as �2[C], where � is the standard deviation inthe permeability of each gridblock. The standard deviation usedwhile generating the initial permeability field (Eq. 1) is usuallymuch higher than the one used while adding the model noise in Eq.2. The correlation length, l, is also treated as a normally distributedstochastic variable. This means that (the noise added to) eachensemble member is generated by first drawing l from a normaldistribution, then generating [C] and finally drawing a permeabil-ity field from a multinormal distribution with zero mean and co-variance matrix �2[C]. To generate the stepwise model noise, thisprocess is repeated, starting by resampling a random correlationlength l.

To generate the noise added to time-dependent variables of themodel, we draw from a normal distribution with zero mean andcovariance matrix [Q]d. In the examples, we use [Q]d��[I], where�≈2×10−16. In generation of the initial ensemble there is no noisein the state variables, as all the ensemble members are started usingthe same equilibrium condition of the reservoir.

The initial model error is due to uncertainty in the permeabilityestimate. In addition, a stepwise model error is added. This is doneboth to account for the fact that we are not able to represent the

permeability distribution using a finite ensemble and to mimic areal estimation process where there are more sources of modelingerror. The specification of both the initial and stepwise modelingerror is a subject of further research.

The analyzed state at time level k is computed by taking intoaccount the measurement vector at time level k. The theory as-sumes that there is a linear relationship between the measurements,d�k , and the states, s�k , expressed by the equation

d�k = �H�s�k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

To take into account the fact that there is a nonlinear relationshipbetween the observed quantities (measurements in the well) andthe state variables, we extend the state vector by one state for eachnonlinear measurement. This brings the correspondence betweenall measurements and the state variables into the form of Eq. 4. Weassume that the measurement noise has a multinormal distributionwith zero mean and covariance matrix [R]k. The index k is includedbecause the covariance matrix of the measurement noise may betime-dependent.

In Ref. 12, it was shown that if the measurement is not treatedas a random variable, the updated ensemble will have correctmean, but too-low, variance. This means that to get consistent errorpropagation in the ensemble Kalman filter, one has to treat theobservations as random variables. This is done by using the actualmeasurement as reference, and adding random noise, reflecting ourassumptions on the measurement noise. This means that the actualmeasurement d�k serves as the reference observation. For eachmember of the ensemble, an observation vector d�k,i is generatedrandomly as

d�k,i = d�k + e�k,io , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where e�mk,i is drawn from a multinormal distribution with zero mean

and covariance [R]k.The error covariance matrix for the state variables of the model

is defined in terms of the true state as the expectation

E��s�kf − s�k

t ��s�kf − s�k

t �T�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

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

s�kt ≈ s�k

f =1

N �i=1

N

s�k,if , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where N is the sample size of the ensemble. With this approxima-tion of the true state, an approximation of a left factor of the errorcovariance matrix of the model is

�L�kf =

1

�N − 1��s�k,1

f − s�kf �, · · · , �s�k,N

f − s�kf �� . . . . . . . . . . . . . . (8)

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

�P�kf = �L�k

f ��L�kf �T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

The expression of the Kalman gain matrix is7

�K�k = �P�kf �H�T ��H��P�k

f �H�T + �R�k�−1. . . . . . . . . . . . . . . . . . . (10)

The analyzed state of each member of the ensemble is computed as

s�k,ia = s�k,i

f + �K�k�d�k,i − �H�s�k,if �. . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

The analyzed error covariance matrix, [P]ak of the model can be

computed along the same lines as [P]fk. Because the updating of the

ensemble is linear, the new estimate of the true state, based on theensemble after the analysis step, is

s�kt ≈ s�a

k = s�kf + �K�k �d�k − �H�s�k

f � . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

and the model error covariance matrix after the analysis step is

�P�ka = ��I� − �K�k�H��P�k

f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

The underlying assumptions behind the filter are that there iszero covariance between the model error and the measurement

67March 2005 SPE Journal

error and that both the model error and measurement error areuncorrelated in time.

In evaluating the quality of the solution we use the root meansquare (RMS) error, defined as

� 1

M �j=1

M

�Xj − Xjt�2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

where M is the number of state variables, Xj is the estimate of statevariable j, and Xt

j is the true value of state variable j.

Example: Simplified Field Model. A simplified 2D field modelhas been constructed by taking out a horizontal layer from a modelof a North Sea field. The dimensions of the simplified model are39×55, but with some inactive blocks. There are 1,931 activegridblocks. The simulation spans a time range of 3,955 days. Thereservoir model is shown in Fig. 1.

Measurements are assimilated at least once a month, but alsowhen new wells are starting production, or when wells are shut in.Altogether, there are 171 points in time when measurements areassimilated. The measurements are assimilated from 14 producingwells. Two of them start from the first day. The reservoir is pro-duced by gas injection, and the producers are located along theoil/water contact. There are four gas injectors. There are no mea-surements from the injectors.

In addition to the 14 producers and 4 injectors, there are somedummy wells included in the simulation to adjust for flow throughthe boundary. The rates of the dummy wells are kept constantwhile generating the data and running the ensemble Kalman filter.

The quantities that are tuned with the Kalman filter are thehorizontal permeability, which is assumed to be the same in the x-and y-direction, together with the dynamic variables, which are thepressure, saturation of water and gas, the solution gas/oil ratio, andthe vapor oil/gas ratio. This means that we have both vaporized oiland dissolved gas.

The “true” model is obtained from a stochastic realization ofthe field. Relative permeabilities and PVT properties were taken

from the original field model. The porosity is shown in Fig. 2. Themeasurements are generated by running the simulator with the“true” permeability, and adding noise to the values obtained.

Injection wells are controlled by historical injection rate. Pro-duction wells are controlled by historical reservoir volume rate.The natural logarithm of the permeability is used in the computa-tions with the Kalman filter. All the data of the restart files fromthe reservoir simulator were stored for each ensemble member, andduring the Kalman filter update, those values corresponding to thestate variables were manipulated according to Eq. 12.

The initial ensemble is generated using a mean correlationlength of 16 gridblocks with a standard deviation of six gridblocks.The initial ln-permeability fields for the Kalman filter are gener-ated using a standard deviation of 0.5. For the model noise (Eq. 2),the standard deviation is set to 5�10−4. The initial mean ofthe permeability of the ensemble is set to 1000 md for allthe gridblocks.

The measurement uncertainties are given in Table 1. We applythe filter to take into account the information gained from fourmeasurements for each of the production wells. The measuredquantities are pressure, oil production rate, water cut, and gas/oil ratio.

In Fig. 3, the development in the estimated permeability isshown together with the “true” permeability and the referencesolution. The reference solution is the mean of the initial ensemble,and because the initial ensemble is based on a model with constantmean permeability of 1000 md, the reference model will have apermeability of approximately 1000 md in all cells. One can ob-serve that after 1,611 days, one obtains a good match of the per-meability field especially close to the producing wells. The loca-tions of the producers are shown with circles. For the permeabili-ties along the boundaries, we observe a slight drifting toward theend of the simulation. Further development of the filter is neededto avoid this effect.

The estimated uncertainty in the ln-permeability is presented inFig. 4. The uncertainty is lowest close to the producers, where themeasurements are obtained. The highest uncertainty is close to theboundary of the reservoir. In general, the uncertainty decreases asmore measurements are assimilated. Close to the boundaries, themodel error added in Eq. 2 dominates the reduction in uncertaintyobtained from measurements at later times.

In Figs. 5 through 8, we present the outcome of the filter forsome of the measured variables, together with the reference solu-tion. We have selected to present results from measurements thathave among them the largest relative deviations between the truthand the solution obtained from the Kalman filter. Notice that thefilter is able to follow the measurements closely.

In Figs. 9 through 12, we show forecasts based on the en-semble mean after 4 days, 819 days, and 3,050 days for the samemeasurements as presented in Fig. 5. As expected, the quality ofthe forecasts is improved as more data are assimilated. It is clearly

Fig. 1—Overview of reservoir model showing wells and ini-tial saturations.

Fig. 2—Porosity of reservoir.


seen that the forecasts improve when more data are included, andthat the forecasts based on data up to 3,050 days are almost perfect.

In Fig. 13, we show how the RMS error defined in Eq. 14evolves. The RMS error for the permeability decreases very rap-idly and then stabilizes. Still, the predictions clearly improve withtime as additional measurements become available. Note also thatwith the exception for RS and RV, the RMS values for the statevariables stabilize relatively fast at quite low levels.

DiscussionThe ensemble Kalman filter technology seems promising for tun-ing permeability and dynamic variables, such as the pressure andsaturations, in reservoir simulator models. The filter was able to

recover the main trends of the permeability field rapidly because ofthe continuous downhole pressure measurements. However, al-though new measurements are added continuously, the error in theestimated permeability increases slightly late in time. The reasonfor this instability is not fully understood.

The initial states of the dynamic variables are determined froman equilibrium state. This fixes the initial state of the reservoir.Therefore, the difference between the estimate obtained from thefilter and the true value of the dynamic variables has to increase inthe beginning. After a while, the differences stabilize.

It is crucial for a successful application of the methodology thatthe ensemble of models correctly reflects the uncertainty at alltimes. Therefore, the generation of the initial ensemble and how to

Fig. 4—Uncertainty in estimated ln-permeabilities. The values are the standard deviation in ln-permeability for each block. Note thatthe plots in the bottom row use the same color scale as those in the upper right plot.

Fig. 3—True and estimated permeabilities. The reference solution is the mean of the initial ensemble (before assimilating anymeasurements). The locations of the producers where the measurements are obtained are shown by circles. In the plot of theestimated permeabilities, only the wells that have been producing before the time of obtaining the estimate are included. All theplots use the same color scale, shown in the upper left plot.


add model noise are important questions that also have to be fur-ther addressed.

Technically, there is no limit with respect to how many modelparameters can be updated, but the effect of adding a large numberof parameters on filter stability, as well as other concerns, has notyet been considered.

Conclusions

An ensemble Kalman filter technology is developed for continuousupdating of reservoir simulation models. The example is shown in2D, but the methodology can easily be extended to 3D and appliedto any existing reservoir simulator.

Fig. 6—Measured values, filter solution, true solution, and the reference solution for the produced oil rate in some wells. The legendis the same as in Fig. 5. The true solution is mostly hidden behind the filter solution. Lower limit for the x-axis corresponds to wellstartup time. Upper limit corresponds to end of history.

Fig. 5—Measured values, filter solution, true solution, and the reference solution for the pressure in some wells. A legend is foundin the lower right plot. The true solution is mostly hidden behind the filter solution. Lower limit for the x-axis corresponds to wellstartup time. Upper limit corresponds to end of history.


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

The methodology is applied to a “semisynthetic” field modelwith 14 producers and four injectors. The ensemble Kalman

filter is able to track the measurements and tune the permeabilityfield, and as more measurements are assimilated the forecastsare improved.

Although the presented results are promising, this technology isnovel to reservoir simulation, and further development of the filteris needed for this application.

Fig. 8—Measured values, filter solution, true solution, and the reference solution for the gas/oil ratio in some wells. The legend isthe same as in Fig. 5. The true solution is mostly hidden behind the filter solution. Lower limit for the x-axis corresponds to wellstartup time. Upper limit corresponds to end of history.

Fig. 7—Measured values, filter solution, true solution, and the reference solution for the water cut in some wells. The legend is thesame as in Fig. 5. The true solution is mostly hidden behind the filter solution. Lower limit for the x-axis corresponds to well startuptime. Upper limit corresponds to end of history.


Nomenclature[C] � correlation matrix

d� � measurement vectore� � stochastic noiseE � expectationf � output from reservoir simulator

[H] � measurement matrixi � index (members of ensemble/gridblock coordinates)

[I] � identity matrixj � index (state variables/gridblock coordinates)k � index (assimilation time)

[K] � Kalman gain matrixl � correlation length

[L] � left factor of covariance matrix of model uncertaintyM � number of statesN � number of ensemble members

[P] � covariance matrix of model uncertainty[Q] � covariance matrix of modeling error[R] � covariance matrix of measurement error

s� � state vector� � standard deviation

�2 � variance

Fig. 10—Forecasted values for some of the oil production rates. The forecasts are based on the mean of the initial ensemble (0days), the forecast after 4 days, after 819 days, after 3,050 days, and the true solution (T). Legends are as in Fig. 9. Lower limit forthe x-axis corresponds to well startup time. Upper limit corresponds to end of history.

Fig. 9—Forecasted values for some of the bottomhole pressures. The forecasts are based on the mean of the initial ensemble (0days), the forecast after 4 days, after 819 days, after 3,050 days, and the true solution (T). Legends are shown in the lower right plot.Lower limit for the x-axis corresponds to well startup time. Upper limit corresponds to end of history.


Subscripts

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

Superscripts

a � analyzed (a posteriori)f � forecasted (a priori)

m � model noiset � true

T � matrix transpose

AcknowledgmentsThis work has been supported financially by ENI-Agip SpA,Norsk Hydro ASA, Statoil, and the Research Council of Norway.

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Fig. 12—Forecasted values for some of the gas/oil ratios. The forecasts are based on the mean of the initial ensemble (0 days), theforecast after 4 days, after 819 days, after 3,050 days, and the true solution (T). Legends are as in Fig. 9. Lower limit for the x-axiscorresponds to well startup time. Upper limit corresponds to end of history.

Fig. 11—Forecasted values for some of the water cuts. The forecasts are based on the mean of the initial ensemble (0 days), theforecast after 4 days, after 819 days, after 3,050 days, and the true solution (T). Legends are as in Fig. 9. Lower limit for the x-axiscorresponds to well startup time. Upper limit corresponds to end of history.


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Geir Nævdal is a chief scientist at RF-Rogaland Research inBergen, Norway. e-mail: [email protected]. He holds a PhDdegree in mathematics from the U. of Trondheim. Liv MeretheJohnsen is a reservoir engineer at Norsk Hydro, and has 4 yearsof experience there as a reservoir engineer and research sci-entist. Before joining Norsk Hydro, she worked for 4 years at theNansen Environmental and Remote Sensing Centre as a re-search scientist. She holds an MSc degree in applied math-ematics from the U. of Bergen. Sigurd Ivar Aanonsen is a prin-cipal researcher and professor at the Centre for IntegratedPetroleum Research, U. of Bergen. He has 18 years of experi-ence with Norsk Hydro as a reservoir engineer and researchscientist. Aanonsen holds a PhD degree in applied mathemat-ics from the U. of Bergen. Erlend H. Vefring is a research directorat RF-Rogaland Research and a professor at the U. of Bergen.He holds a PhD degree in applied mathematics from the U.of Bergen.

Fig. 13—RMS error between estimated and true variables (solid line) and between the reference solution and the true variables(dashed). PERMX is the ln-permeability.


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SPE-84372-PA