Real-Time Reservoir Model Updating Using Ensemble Kalman Filter Xian-Huan Wen, SPE and Wen. H. Chen, SPE, ChevronTexaco Energy Technology Company

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1 SPE Real-Time Reservoir Model Updating Using Ensemble Kalman Filter Xian-Huan Wen, SPE and Wen. H. Chen, SPE, ChevronTexaco Energy Technology Company Copyright 2004, Society o Petroleum Engineers Inc. This paper was prepared or presentation at the 2005 SPE Reservoir Simulation Symposium held in Houston, Texas U.S.A., 31 January February This paper was selected or presentation by an SPE Program Committee ollowing review o inormation contained in a proposal submitted by the author(s). Contents o the paper, as presented, have not been reviewed by the Society o Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily relect any position o the Society o Petroleum Engineers, its oicers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees o the Society o Petroleum Engineers. Electronic reproduction, distribution, or storage o any part o this paper or commercial purposes without the written consent o the Society o Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal o not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acnowledgment o where and by whom the paper was presented. Write Librarian, SPE, P.O. Box , Richardson, TX , U.S.A., ax Abstract The ensemble Kalman Filter technique (EnKF) has been reported to be very eicient or real-time updating o reservoir models to match the most current production data. Using EnKF, an ensemble o reservoir models assimilating the most current observations o production data are always available. Thus the estimations o reservoir model parameters, and their associated uncertainty, as well as the orecasts are always upto-date. In this paper, we apply the EnKF or continuously updating an ensemble o permeability models to match realtime multiphase production data. We improve the previous EnKF by resolving the low equations ater Kalman ilter updating so that the updated static and dynamic parameters are always consistent. By doing so, we show that the production data are also better matched or some cases. We investigate the sensitivity o using dierent number o realizations in the EnKF. Our results show that a relatively large number o realizations are needed to obtain stable results, particularly or the reliable assessment o uncertainty. The sensitivity o using dierent covariance unctions is also investigated. The eiciency and robustness o the EnKF is clearly demonstrated using an example. By assimilating more production data, new eatures o heterogeneity in the reservoir model can be revealed with reduced uncertainty, resulting in more accurate predictions. Introduction The reliability o reservoir models increases as more data are included in their construction. Traditionally, static (hard and sot) data, such as geological, geophysical, and well log/core data are incorporated into reservoir geological models through conditional geostatistical simulation 1. Dynamic production data, such as historical measurements o reservoir production, account or the majority o reservoir data collected during the production phase. These data are directly related to the recovery process and to the response variables that orm the basis or reservoir management decisions. Incorporation o dynamic data is typically done through a history matching process. Traditionally, history matching adjusts model variables (e.g., permeability, porosity, and transmissibility, etc.) so that the low simulation results using the adjusted parameters match the observations. It requires repeated low simulations. Both manual and (semi)automatic history matching processes are available in the industry Automatic history matching is usually ormulated in the orm o a minimization problem in which the mismatch between measurements and computed values is minimized Gradient-based methods are widely employed or such minimization problems, which require the computation o sensitivity coeicients In the recent decade, automatic history matching has been a very active research area with signiicant progress reported 17. However, most approaches are either limited to small and simple reservoir models or are computationally very intensive. Under the ramewor o traditional history matching, the assessment o uncertainty is usually through a repeated history matching process with dierent initial models, which maes the process even more CPU demanding. In addition, the traditional history matching methods are not designed in such a ashion that allows or continuous model updating. When new production data are available and are required to be incorporated, the history matching process has to be repeated using all measured data. These limit the eiciency and applicability o the traditional automatic history matching techniques. On the other hand, during recent years, more and more permanent sensors are being deployed or monitoring pressure, temperature, or low rates. The data output requency in this case is very high. It has become important to incorporate the data as soon as they are available so that the reservoir model is always up-to-date. Traditional history matching is not suitable or such a purpose because o the heavy computational burden and the high data sampling requency. A new ind o history matching method that can use all recorded data or ast and continuous model updating is needed. The Kalman ilter was originally developed to continuously update the states o linear systems to account or available measurements. When the system is non-linear, the

2 2 X.-H. Wen and W. H. Chen SPE extended Kalman ilter was proposed to linearize the nonlinear system. For very large models or highly non-linear systems, the extended Kalman ilter ails. The ensemble Kalman ilter (EnKF) was then introduced to overcome some o the problems o the extended Kalman ilter 18. Particularly, instead o evaluating the necessary statistics (e.g., correlation between model parameters and responses) based on linear assumptions, EnKF uses an ensemble o model representations rom which all necessary statistics can be directly computed. Base on this, EnKF is a Monte-Carlo approach. The EnKF has been widely used in the areas o weather orecasting, oceanography and hydrology In these applications, only dynamic variables are tuned. It has recently been introduced in the Petroleum industry or continuous updating o reservoir models where both static and dynamic variables are simultaneously tuned to assimilate new measurements. Nævdal et al. 23 used the ensemble Kalman ilter to update static parameters in near-well reservoir models by tuning the permeability ield. This approach was later urther developed to update 2-D three-phase reservoir models by continuously adjusting both the permeability ield and saturation and pressure ields at each assimilation step 24. Assimilation happened at least once a month and also when new wells started to produce or wells were shut in. Gu and Oliver used the ensemble Kalman ilter to update porosity and permeability ields, as well as the saturation and pressure ields, then applied it to match 3-phase production data at wells rom the 3-D PUNQ-S3 reservoir model. Furthermore, Brouwer et al. 27 used the combination o EnKF or continuous model updating with an automated adjoint-based water lood optimization to optimize water looding strategy. Results rom the previous studies have shown that the EnKF is very eicient and robust. However, they all indicated that the estimate o the permeability ield got worse at late time, which is oten reerred to as ilter divergence 28. In this paper, we present a modiied version o the EnKF or continuously updating an ensemble o permeability models to match real-time multiphase production data. We improve the previously reported EnKF by adding an option o running reservoir simulation ater updating using the most recent updated static model parameters so that the updated static parameters and dynamic parameters are always consistent. We also investigate some o the important issues or the EnKF including the number o realizations in the ensemble, and the sensitivity o covariance model. The outline o this paper is as ollows. We irst briely describe the methodology o the EnKF and the implementation in connection with a reservoir simulator. Then we present a 2-D synthetic example o using the EnKF to continuously update the permeability model assimilating realtime observations o multiphase production data at wells. Next, we discuss some o the important issues, ollowed by the summary and conclusions. Ensemble Kalman Filter The EnKF consists o three processes or each time step: orecast based on current state variables (i.e. solve low equations with current static and dynamic parameters), data assimilation (computation o Kalman gain), and updating o state variables. The evolution o dynamic variables is dictated by the low equations. State variables include three types o parameters: static parameters (e.g., permeability and porosity ields that are traditionally called static because they do not vary with time. However, in the EnKF approach, static parameters are updated with time and thus can change with time. We use this notion or the convenience o traditional concepts), dynamic parameters (e.g., pressure and phase saturations o entire model that are usually solutions o the low equations), and production data (e.g., well production rate, bottom-hole pressure, water cut, etc. that are usually measured at wells). The ensemble o state variables is modeled by multiple realizations. Thus, we have ms y, j = m d (1) d where time t. j, j y, is the j th ensemble member o the state vector at m s and m d are static and dynamic vectors, and d is the production data vector. In this paper, m s is the permeability at each cell o the reservoir model with dimension o N being the total number o active cells, md includes pressure and water saturation (two-phase model) at each cell (with dimension o 2N), and d includes bottomhole pressure, oil production rate and water production rate at wells with dimension o N d,. The dimension o state vector is N y,. which can change with time t to account or dierent amount o production data at dierent time. The step-by-step process o a typical EnKF is described as ollows (see Fig. 1). The ilter is initialized by generating initial ensembles o static and dynamic vectors. There is no production data available at the starting time (t 0 ). In this paper, we use geostatistical method to generate multiple (200) realizations o the permeability ield with given statistical parameters (histogram and variogram) to represent the initial uncertainty in the permeability model beore any production data are available. Initial dynamic variables (i.e., initial pressure and saturations) are assumed nown without uncertainty. Thus they are the same or each realization. I, however, initial variables are also uncertain, they should be represented by ensembles. The orecast step calls a reservoir simulator or each o the realizations until the next point in time where new measurements o production data are available and to be assimilated (e.g., t 1 ). The state vector ater running the orecasting is denoted by y = t1, j (note the values o static variables m s at the initial and orecast steps are the same). Forecasting creates the ensemble o new dynamic and production data at the given time

3 SPE Real-Time Reservoir Model Updating Usign Ensemble Kalman Filter 3 consistent with the initial static parameters. We can then compute the Kalman gain as: (2) where K is the Kalman gain or time t. H is a matrix operator that relates the state vector to the production data. Because the production data are part o the state vector as in Eq. (1), H is in the orm o H = [ 0 I], where 0 is a N d, ( y, d, N d Nd, N N ) matrix with all 0 s as its entries; I is a, identity matrix. The superscript denotes orecast, meaning that the values are output rom the simulator beore Kalman ilter updating. C, is the covariance matrix o d N N, production data with dimension o d, d and is diagonal in this paper since we assume the production data errors are independent. C y, is the covariance matrix or the state variables at time t that could be estimated rom the ensemble o orecasted results ( ) using the standard statistical method: y, j 1 T C (3) y, = ( Y Y )( Y Y ) Ne 1 where Y is the ensemble o orecasted state vector at time t with dimension o N N N is the y, e ( e number o realizations in the ensemble). is the mean o the state variables which is a vector with dimension o N y,. With K and production data at the assimilating time step ( d ), the state vector is then updated as: y K u, j T T 1 = Cy, H ( H Cy, H + Cd, ) = y + K d H y ) (4), j (, j, j Note that random perturbations are added to the observed production data ( d ) to create an ensemble o production data set with d, j being the j th ensemble member. Burgers et al. 29 shown that the variability among the updated ensemble members is too small i no random noise is added to the production data. Eq. (4) has apparent physical meaning: the second part o the second term on the right hand side is the dierence o simulated and observed production data; the larger this dierence, the larger update will be applied to the initial state vector. I the simulated production data at a given realization is equal to the observed data, no update will need to this model. The covariance matrix ater updating can be computed as: u C y, = ( I KH ) Cy, Y (5) With this updating, the state vector o each realization in the ensemble is considered to relect the most current production data ( d ) and we can proceed to the next time step where new production data are available or assimilation. The state variables are advanced in time as: y, j u = F y ) j = 1, 2, N e (6) ( 1, j where F is the reservoir simulator. In this paper, we use our in-house reservoir simulator. From Eqs. (4) and (5), we can see that the updating o the ensemble is linear and there is an underlying assumption that the model error and production data error are independent. Also, both model and production data errors are uncorrelated in time. The overall wor-low o EnKF is shown in Fig. 1 ollowing the solid arrows. We can see that production data are incorporated into reservoir models sequentially in time as they become available and the ensemble o reservoir models are evolving with time representing the assimilation o measurements at the given time. When new measurements o production data are acquired, we simply orward simulate the low using the most current state vector to the time at which new production data are collected and perorm the above analysis to update the state vector in order to relect the new data. Each assimilation represents some degree o increment o quality to the estimation o reservoir model. The degree o this increment depends on how much inormation the new measured data is carrying. Thus, there is no need to start the process all over again rom the original starting time in order to incorporate the new acquired data, whereas, in traditional history matching, static parameters are treated as static (not varying with time) and all production data are matched simultaneously using one set o static parameters. When there are new measurements needed to be matched, the entire history matching has to be repeated using all data. The advantage o using EnKF is obvious, especially, when the requency o data is airly high as, or example, the data rom permanent sensors. The EnKF can be built upon any reservoir simulator because it just requires the outputs o the simulator. The simulator acts as a blac box in the process o EnKF. Thus the coding o the EnKF is signiicantly simpler than traditional gradient-based history matching methods where complicated coding o sensitivity calculations is required or dierent simulators and access to the source code o the simulator is needed 15. Another advantage o EnKF is that it provides an ensemble o N e reservoir models all o which match up-to-date production data with the computer time o approximately N e low simulations (the CPU time or data assimilation is very small compared to the low simulation). This is well-suited or uncertainty analysis when multiple reservoir models are needed. With traditional history matching, however, we need to repeat the CPU intensive history matching process with dierent initial models to create multiple models, which is very ineasible. Furthermore, the EnKF is well-suited or

4 4 X.-H. Wen and W. H. Chen SPE parallel computation since the time evolution o ensemble reservoir models are completely independent. However, one potential problem arises by using the EnKF to simply update dynamic variables as in Eq. (4) without the constraint o the low equations: the updated dynamic variables m d (e.g., pressure and saturations) may not be physically meaningul and may be inconsistent with the updated static variables m s (e.g., permeability) o the same time. This is due to the act that Kalman updating is linear, whereas the low equations are non-linear. Some methods have been proposed to remedy this 25, but they are not very eective. In this paper, we propose to add an additional component to the EnKF called conirming to ensure that the updated dynamic and static variables are always consistent. The idea is the ollowing (ollow the thic solid arrows and dotted arrows in Fig. 1): start rom t 0, orecast to the next time (t 1 ), then compute Kalman gain and update state variables as in the previously described process (i.e., ollow the thic solid arrows), then tae only the newly updated static parameters m s (i.e., permeability) and run the low simulation rom only the current t 0 to the next time t 1 again (i.e., the conirming step, ollow the dotted arrows in Fig. 1). The newly low simulated dynamic vector m d replace the Kalman ilter updated dynamic vector and are used as the starting/initial vector or the next assimilation step. By doing so, we ensure that the updated static and dynamic parameters are always consistent with the low equations. The cost is a doubling o the CPU time compared to the no-conorming EnKF. We will show later that production data are better matched by using this additional conirming step. Also it is possible to iterate the conirming step to chec the matching o production data. In the next section, we will demonstrate the ability and eiciency o the proposed EnKF or continuously updating reservoir models using a 2-D synthetic example. Example Fig 2a shows a 2-D geostatistical reerence ield (50x50x1 grid with cell size 20 eet x 20 eet x 2 eet). The model is generated using the Sequential Gaussian Simulation method 1. The ln() has a Gaussian histogram with mean and variance o 6.0 and 3.0, respectively. The unit o permeability is milli Darcy. The variogram is spherical with range o 200 eet and 40 eet in the direction o 45 degrees and 135 degrees, respectively. We assume an injection well (I) at the center o the model with 4 production wells (P1 to P4) at the 4 corners. The main eatures o this reerence ield are: (1) a high permeability zone and a low permeability zone in the middle o the ield, (2) high interconnectivity between well I and well P1, (3) low interconnectivity between well I and wells P3 and P4. This reerence ield is considered as the true model, and our goal is to reconstruct reservoir models, based on real-time production data, which are as close to the true ield as possible. The reservoir is initially saturated with oil with constant initial pressure o 6000 psi at the top. The injection well has constant injection rate o 700 STB/day with a maximum bottom-hole pressure (BHP) control o psi. All producers are producing a constant total volume o 200 STB/day with minimum BHP control o 4000 psi. The mobility ratio o water and oil is 10 and standard quadratic relative permeability curves are used with zero residual saturation or oil and water. Compressibility and capillary pressure are ignored. Flow simulation is run to 720 days and results o BHP o each well, as well as oil production rates (OPR) and water production rates (WPR) at producing wells are shown in Figs. 2b-d. Note the ast water breathrough and high water production rates at well P1, whereas or P3, water breathrough is very late with small WPR and the BHP o this well drops to minimum control right ater production, due to the low permeability around this well as well as low connectivity between this well and the injector. The reerence permeability ield, as well as the simulated dynamic data (BHP, OPR and WPR) are considered as truth and we assume the measurements o BHP, OPR and WPR at wells are available every 30 days up to 300 days and they are directly read rom the true data. The standard deviations o measurement errors are 3 psi, 1 STB, and 2 STB or BHP, OPR and WPR, respectively. Gaussian random errors are added to the perect true production data to create a noisy data set. A perturbation vector with the same variances as measurement errors is then added to the noisy data to create an ensemble o production data 26,29. An initial ensemble o permeability models are generated using the Sequential Gaussian Simulation method with the same histogram and variogram as the reerence ield. We assume that there is no hard permeability available, thus all initial models are unconditional. Conditional simulation can be used when there are hard and/or sot data o permeability and they can be preserved during the EnKF updating. Other parameters (porosity = 0.2, relative permeability curves, initial pressure = 6000 psi, and initial water saturations = 0.0) are assumed nown without uncertainty. Note that i the initial parameters or boundary conditions o a reservoir are uncertain, this can also be accounted or by using an ensemble o representations as well. This will be discussed in a uture paper. The ensemble o 200 initial permeability models are input or the EnKF and are updated at every 30 days assimilating the observed production data (BHP, OPR, and WPR) at the given times. Fig. 3 shows the mean/averaged permeability ields (i.e., the estimation) and the associated variance ields (uncertainty) computed rom the ensemble at 0, 30, 60, and 300 days, respectively. The variance ield is the same as the diagonal terms in the updated covariance matrix rom Eq. (5). The conirming step is used during the EnKF updating. Compared to the true model (Fig. 2a), we can see that: (1) at the starting time when there is no production data available, the averaged model and its variance ield are eatureless with constant values close to the mean and variance o the input histogram (irst row o Fig. 3); (2) at the irst assimilation step (30 days), most spatial variation eatures in the reerence model are captured by the averaged model with reduced

5 SPE Real-Time Reservoir Model Updating Usign Ensemble Kalman Filter 5 uncertainty around the wells and between well areas (second row o Fig. 3); (3) as time proceeds and more production data are assimilated, improved averaged permeability models are obtained with urther reduced uncertainty (third and orth rows o Fig. 3); (4) at the later time, the averaged permeability ields and variance ields become closer and closer between the dierent assimilation steps (e.g., at 120 and 300 days) indicating that the production data at the later time carry less and less useul inormation on the reservoir heterogeneity compared to the early time data. This indicates the importance o assimilating early time production data or ast recognition o important reservoir heterogeneity. Any production data that capture signiicant low behavior changes in reservoir (such as adding new wells, well shut-in, or conversion o a producing well to an injection well, etc.) should be assimilated as early as possible. As mentioned beore, by using the EnKF, an ensemble o reservoir models that are consistent with the up-to-date production data are available or predictions o uture perormance. This provides the possibility o assessing uncertainty in the orecasts. Fig. 4 shows predictions o production data at some selected wells using the reservoir models updated at dierent time (0, 30, 60, 120, and 300 days). Only results rom the irst 10 models are shown. Flow simulations are run rom the starting time until 720 days or each case. It is obvious that at the starting time, when no production data are available, the well perormance predicted by the initial models are neither accurate (deviated largely rom the true results) nor precise (with large uncertainty). As more and more production data are available and assimilated at later time, the predictions become more and more accurate (close to the true results) and precise (less spreading among realizations), particularly, or BHP at P1 and WPR at P3. Alternatively, predictions can be perormed on a single averaged model (considered as the best estimation) i there is no need to estimate the uncertainty or the predictions. Fig. 5 shows predictions o well perormance at selected wells using the mean/averaged updated models at dierent times (0, 30, 60, 120, and 300 days). We can see that, by using the averaged models assimilating only early time production data (e.g., 0, 30, and 60 days), the predicted well perormances, although improved, still signiicantly deviate rom the true answers (particularly or BHP) indicating that assimilating production data up to 60 days is not suicient to capture the spatial heterogeneity eatures in the reerence model that are important to the underlying low/well pattern. However, when the production data at 300 days are assimilated, the updated averaged model provides predictions that are very close to the reerence results. This indicates that the quality o updated reservoir model is improved gradually with time as more production data are assimilated. Although the changes to the permeability models are small at late time, a certain amount o useul inormation is still captured. Fig. 6 shows the evolution o a single realization at dierent times. The associated well perormance predictions are presented in Fig. 7. We can see that, initially, the permeability values in the area between wells I and P3 are too high compared to the true model, whereas in the area between wells I and P2, permeabilities are too low. The well perormances predicted by this model are signiicantly dierent rom the true answers. At 30 days, when irst available production data are assimilated, the updated model displays the spatial heterogeneity eatures that are signiicantly closer to the reerence model than the initial model. However, the well perormances predicted by this model are still considerably dierent rom the true results, particularly or BHP at P4 and WPR at P3 (Fig. 7). At later time when more and more production data are assimilated, well perormances predicted by the updated model are closer and closer to the true results indicating more and more heterogeneity eatures in the reerence ield are revealed by the production data and the successive assimilation o useul inormation carried by the production data into the reservoir model by the EnKF. Discussions In this section, we investigate some important issues in the EnKF including the number o realizations in the ensemble, covariance unction, and the importance o using the conirming step as discussed previously. Size o Ensemble. Previous studies used a small number o realizations (varying rom 40 to 100) in the EnKF In order to investigate how many realizations are needed in the EnKF to reliably represent the uncertainty o the model, as well as the uncertainty in predictions, we perorm the EnKF with the same data set as in the previous section using 50, 100, 200 and 400 realizations in the ensemble. Fig. 8 shows the mean (average) and variance ields o permeability updated at 120 days using dierent number o realizations. Clearly, with small ensemble size (50 and 100), the uncertainty is underestimated, although the averaged model can suiciently represent important spatial eatures in the reerence ield. Using 200 realizations seems adequate to represent the uncertainty in the permeability model. We also perorm low simulation on each o the realizations updated by using dierent ensemble sizes. Fig. 9 shows the BHP at P1 computed rom 50 realizations updated at 300 days by using dierent ensemble sizes. They all can closely match the production data. But the spreading o well perormances among realizations is smaller when smaller ensemble size is used. More importantly, the prediction range at late time does not cover the true results when only 50 or 100 realizations are used in the ensemble, indicating bias in the results (Figs. 9a and b). When larger ensemble sizes are used, the prediction results are not biased (Figs. 9c and d). Thus, we can conclude that a relatively large ensemble size is required in order to adequately represent the uncertainty o model parameters. I, however, the prime goal o the EnKF is to create reservoir models that match the up-to-date production data only, with less emphasis on the accurate representation o uncertainty, the use o small ensemble size may be justiied. Liu and Oliver 30 investigated the use o multiple sets o EnKF updating each o which used small ensemble size (40). They ound that the predictions o well perormances rom the dierent sets o models could vary signiicantly with dierent degree o accuracy and uncertainty. This also indicates the

6 6 X.-H. Wen and W. H. Chen SPE insuicient number o realizations in the ensemble. Another observation is that the ilter divergence happens earlier when smaller ensemble size is used. Covariance Function. In many practices, we usually do not have access to the accurate covariance unction o the model parameter errors. To investigate the sensitivity o using dierent covariance model on the EnKF results, we generated the initial ensemble permeability models using an isotropic variogram with correlation length o 100 eet (note that the variogram or the reerence ield is anisotropic with correlation lengths o 200 eet and 40 eet in the directions o 45 degree and 135 degree, respectively). The evolution o averaged permeability models and the associated variance ields rom 200 realizations at dierent assimilation time (30, 60, 120, and 300 days) are shown in Fig. 10. Visually, the averaged model seems to have dierent spatial patterns rom the results shown in Fig. 3, which is expected because dierent variogram models are used. Nevertheless, results rom both igures closely reproduce the most eatures in the reerence ield that are important or low (e.g., high permeability between I and P1, and low permeability between I and P4, etc.). The uncertainty is also reduced by the similar degree with the similar ashion compared to Fig. 3. The low prediction results are also very similar to those shown in Fig. 4 where an accurate variogram model is used. This demonstrates that the covariance model is not critically important in the EnKF in order to capture the important spatial variation eatures. As long as these eatures are relected in the production data, they can be revealed in the updated model eventually. Conirming vs. No-conirming. We next investigated improvement o the updated results by using the additional conirming step. Fig. 11 presents the comparisons o pressure and water saturation ields at 30 days or a given realization by using conirming and no-conirming updating. The irst row shows the reerence permeability (i.e., m s ) and the corresponding pressure and water saturation ields (i.e., m d ). The second row presents the initial permeability and its associated pressure and water saturation ield beore assimilating any production data. Notice the apparent dierences o this model and its dynamic responses rom the reerence ield. The third row shows the updated permeability model, as well as the updated pressure and water saturation ields based on the Kalman gain using the production data at 30 days (i.e., Eqs. (2) and (4)). We can see that, compared to the initial results, the updated results (both m s and m d ) are closer to the true results with still considerable dierence. But the updated permeability, pressure and water saturation results are not consistent with the low equations. The ourth row shows the recomputed pressure and water saturation ields using the Kalman ilter updated permeability model (i.e., Fig. 11g), i.e., they have been conirmed to the updated permeability model. It can be clearly seen that the conirmed pressure and water saturation ields are much closer to the reerence results compared to the Kalman ilter updated results (third row). More importantly, all static and dynamic variables are now consistent and constrained to the low equations. Fig. 12 shows the same comparisons as in Fig. 11, but at 120 days. Again, the irst row is the reerence model with the pressure and water saturation ields at 120 days. The second row shows the initial permeability model (i.e., at the previous assimilating time o 90 days), as well as the resulting dynamic data (pressure and water saturation) simulated based on the initial permeability model. Compared to the previous igure, we can see that the dierences between the dynamic data and the reerence results are much smaller or this case because this initial permeability model has already assimilated production data observed at the three previous time steps (at 30, 60, and 90 days) and is already quite close to the reerence model (compare Figs. 12a and d). By assimilating production data at 120 days, these variables are then updated and shown in the third row o Fig. 12. Note that the changes applied to the permeability and water saturation are very small compared to their original values shown in the second row, whereas, large changes occur or the pressure with the updated pressure values signiicantly underestimating the reerence results. This demonstrates the clear inconsistency between updated permeability (m s ) and updated pressure (m d ). Through the conirming run o low simulation using the updated permeability model, the resulting pressure and water saturation are given in the last row o Fig. 12. The conirming pressure and water saturation are now consistent with the updated permeability model, and they are closer to the reerence results compared to the initial and Kalman ilter updated results (particularly or pressure). They now replace the Kalman ilter updated dynamic data (shown in the third row) and will be used as initial state variables or the next assimilating step. Finally, we present the predictions o well perormance using the permeability models updated by conirming or noconirming EnKF. Most models updated by both conirming and no-conirming EnKF provide similar predictions, with a small number o exceptions. Fig. 13 shows one realization o the permeability model updated at 120 days using the conirming (Fig. 13a) and the no-conirming (Fig. 13b) EnKF. Although they loo very close, the dierence is clearly shown in Fig. 13c (conirming values no-conirming values). The predictions o BHP at well P1 using the dierent permeability models are plotted in Fig. 13c. The permeability model updated by the no-conirming EnKF yield the BHP prediction that is dramatically larger than the true result, while the model updated by the conirmed EnKF predicts the BHP very close to the true result. This clearly demonstrates the advantages o using conirming EnKF. Our investigation also indicates that the dierences between the Kalman ilter updated dynamic data and the conirming dynamic data become smaller and smaller at the late time when the production data carry less and less new and valuable inormation on reservoir heterogeneity. Thus, we can switch the EnKF bac to the no-conirming option to save some computation time. Whenever new wells are added, or with signiicant well condition changes (e.g., shut-in or conversion o producer to injector, etc.), we switch bac to the conirming option.

7 SPE Real-Time Reservoir Model Updating Usign Ensemble Kalman Filter 7 Adding New Wells During Assimilation. Results presented so ar in this paper are obtained using the same wells with the same production controls. Our results show that the value o production data is diminishing with time when suicient amount o useul inormation carried by the same type o production data rom the same wells is assimilated. This is relected by the small changes applied to the state vector at late assimilation steps with stable predictions. In actual practice, new wells are usually drilled and added to the production during the production phase, or well conditions are requently varying, which may cause signiicant low behavior changes in the reservoir. The new production data that capture such changes carry new inormation on the underlying reservoir heterogeneity that the previous data may not carry. Assimilating the new production data can urther improve the estimation quality o reservoir model with urther reduction o uncertainty. Besides the production data rom the 5 wells used in the previous results, we now assume that 4 more production wells are drilled at 120 days and start production at constant total volume rate o 100 STB/day with minimum BHP control o 4000 psi. Also the injection rate at the injector is increased to 900 STB/day. Production data at all wells are measured at every 30 days and they are assimilated to the reservoir models with EnKF. Fig. 14 shows the evolution o average and variance ields rom 200 realizations updated at dierent time (120, 150, 180, and 600 days, respectively). Note that beore 120 days, new wells are not added and results are the same as in Fig. 3. When the production data at new wells are assimilated, new eatures o heterogeneity in the reerence ield are revealed with urther reduction o uncertainty at new well locations, as well as at the between well areas. Figs. 15 and 16 show the matching o well perormance o 10 realizations updated at 600 days using production data o the original 5 well only or 9 wells (5 wells with data rom 0 to 300 days and 4 additional wells with data rom 120 to 300 days). We can see that production data at the new wells are not well matched when their data are not assimilated (Fig. 15). This indicates that some heterogeneity eatures impacting low behavior at new wells are not captured when their production data are not assimilated. All realizations can closely reproduce the production data at all wells when all production data at all wells are incorporated (Fig. 16). Summary and Conclusions The ensemble Kalman ilter has been implemented with our in-house reservoir simulator to continuously update an ensemble o reservoir models assimilating real-time production data. A conirming option is added to ensure the updated static and dynamic variables are always consistent. This avoids the possible nonphysical values or the updated dynamic data. A 2-D synthetic example is used to illustrate the application o the newly developed EnKF methodology to update permeability models assimilating successively realtime multi-phase production data. Sensitivities o using dierent ensemble sizes and dierent covariance models are investigated. The advantages o using conirming EnKF are also discussed. The ollowing conclusions can be drawn rom this study: The EnKF is very eicient and robust or real-time updating o reservoir models to conirm the newly collected production data. The total CPU time or creating N e realizations o reservoir models that match the latest production data is about the cost o running N e reservoir simulations. An ensemble o reservoir models that are consistent with the up-to-date production data are always available or predictions o uture perormance with assessment o uncertainty. A relatively large ensemble size is required in order to accurately estimate the uncertainty o model parameters. I, however, the goal o EnKF is to create reservoir models that match the up-to-date production data only, with less emphasis on the accurate representation o uncertainty, the use o small ensemble size may be justiied. The covariance model used in the EnKF is not critically important in order to capture the important spatial variation eatures. As long as these eatures are relected in the production data, they can be revealed in the updated models eventually. The use o conirming EnKF ensures the updated static and dynamic variables are always consistent, avoiding the possible nonphysical values or the updated dynamic data. The production data are better matched or some cases using the conirming EnKF. In general, by assimilating more production data, new eatures o heterogeneity in the reservoir model can be revealed with reduced uncertainty, resulting in more accurate predictions. However, i the available production data are the same type o data and rom the same wells or a long time, the value o production data is diminishing with time resulting in less and less updating at the later time. Acnowledgments We than Alan Bernath and Xundan Shi o ChevronTexaco, ETC or their assistances and many useul discussions. We also than Pro. D. S. Oliver o University o Olamoma or useul discussions. Nomenclatures C d = Covariance o production data error C = Covariance matrix o state vector y c m, l = Element in C y d F H I K ln() m s m d N = Production data vector = Forward low simulator = Matrix operator that relates state vector to production data in state vector = Identity matrix = Kalman gain = Natural logarithm o permeability = Static variables = Dynamic variables = Total number o active cells in reservoir model

8 8 X.-H. Wen and W. H. Chen SPE N d N e N y t Y Y y = Dimension o production data vector = Number o realizations in ensemble = Dimension o state vector = Time = Ensemble o state vector = Mean o the state vector = State vector Subscript = Time index j = Ensemble member index Superscript = Forecast T = Transpose u = Updated Reerences 1. Deutsch, C.V. and Journel, A.G.: GSLIB: Geostatistical Sotware Library and User s Guide. 2 nd edition, Oxord University Press, New Yor, 369pp, Chen, W. H. et al.: A new algorithm or automatic history matching, S PEJ., , He, N., Reynolds, A. C., and Oliver, D. S.: Three-dimensional reservoir description rom multiwell pressure data and prior inormation, paper SPE presented at the 1996 SPE Annual Technical Conerence and Exhibition, Denver, CO, 6-9 October, Landa, J.L. and Horne, R.N.: A Procedure to Integrate Well Test Data, Reservoir Perormance History and 4-D Seismic Inormation into a Reservoir Description, paper SPE presented at the 1997 SPE Annual Technical Conerence and Exhibition, San Antonio, TX, 5-8 October, Millien, W.J. and Emanuel, A.: History Matching Finite Dierence Models with 3D Streamlines, paper SPE presented at the 1998 SPE Annual Technical Conerence and Exhibition, New Orleans, LA, September, Vasco, D. W., Yoon, S., and A. Datta-Gupta, Integrating Dynamic Data Into High-Resolution Reservoir Models Using Streamline-Based Analytical Sensitivity Coeicients, SPE paper presented at the 1998 SPE Annual Technical Conerence and Exhibition, New Orleans, LA, September, Wen, X.-H., Deutsch, C.V. and Cullic, A.S.: High Resolution Reservoir Models Integrating Multiple-Well Production Data, SPEJ, , December, Wen, X.-H, Deutsch, C.V. and Cullic, A.S.: Integrating Pressure and Fractional Flow Data in Reservoir Modeling With Fast Streamline-Based Inverse Method, paper SPE presented at the 1998 SPE Annual Technical Conerence and Exhibition, New Orleans, LA, September, Roggero, F. and Hu, L. Y.: Gradual Deormation o Continuous Geostatistical Models or History Matching, paper SPE 49004, presented at 1998 SPE Annual Technical Conerence and Exhibition, New Orleans, LA, September, Agarwal, A. and Blunt, M.J.: Streamline-Based Method with Full-Physics Forward Simulation or History-Matching Perormance Data o a North Sea Field, SPE Journal, 8(2), June, Caers, J.: Geostatistical history matching under training image based geostatistical model constraints, SPEJ, , Cheng, H., Wen, X.-H., Millien, W. J., and Datta-Gupta, A.: Field experiences with assisted and automatic history matching using streamline models, paper SPE presented at the 2004 SPE Annual Technical Conerence and Exhibition, Houston, TX, September, Tarantola, H.: Inverse Problem Theory: Methods or Data Fitting and Model Parameter Estimation. Elsevier, Amsterdam, Netherlands, 613pp, Sun N.-Z.: Inverse Problem in Groundwater Modeling. Kluwer Academic Publishers, Boston, 337pp, Li, R., Reynolds, A. C. and Oliver, D. S.: Sensitivity coeicients ro three-phase history matching, Journal o Canadian Petroleum Technology, 42(4), , Wen, X. H., Deutsch, C. V. and Cullic, A. S.: Inversion o Dynamic Production Data or Permeability: Fast Streamline- Based Computation o Sensitivity Coeicients o Fractional Flow Rate, J. o Hydrology, vol. 281, , Wen, X.-H, Deutsch, C.V. and. Cullic, A.S.: A review o Current Approaches to Integrate Flow Production Data in Geological Modeling, Report 10, Stanord Center or Reservoir Forecasting, Stanord, CA, Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to orecast error statistics, Monthly Weather Review, 127(12), , Houteamer, P. L. and Mitchell, H. L.: Data assimilation using an ensemble Kalman ilter technique, Monthly Weather Review, 126(3), , Reichle, R. H., McLaughlin, D. B., and Entehabi, D.: Hydrologic data assimilation with the ensemble Kalman ilter, Monthly Weather Review, 130(1) , Margulis, S. A., McLaughlin, D., Entehabi, D., and Dunne, S.: Land data assimilation and estimation o soil moisture using measurements rom the Southern Great Plains 1997 Field Experiment, Water resources Research, 38(12), Enevsen, G.: The ensemble Kalman ilter: Theoretical ormulation and practical implementation, Ocean Dynamics, 53(4), , Nævdal, G., Mannseth, T., and Vering, E. H.: Near-well reservoir monitoring through ensemble Kalman ilter, paper SPE presented at the SPE/DOE Improved Oil Recovery Symposium, April, Nævdal, G., Johnsen, L. M., Aanonsen, S. I., and Vering, E. H.: Reservoir monitoring and continuous model updating using ensemble Kalman ilter, paper SPE presented at the 2003 SPE Annual Technical Conerence and Exhibition, Denver, CO, 5-8 October, Gu, Y. and Oliver, D. S.: The ensemble Kalman ilter or continuous updating o reservoir simulation models, Computational Geosciences, in press, Gu, Y. and Oliver, D. 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9 SPE Real-Time Reservoir Model Updating Usign Ensemble Kalman Filter 9 Figure 1: Description o overall wor-low o the EnKF Figure 2: The Reerence permeability ield and production data. Figure 3: The evolution o mean/average and variance ields updated by EnKF at dierent time computed rom 200 realizations.

10 10 X.-H. Wen and W. H. Chen SPE Figure 6: The evolution o one permeability realization at dierent time. Figure 4: Well production data simulated rom 10 realizations updated by EnKF at dierent time. Results rom the reerence model are in red. Figure 7: Well production data simulated by using a single realization updated at dierent time as shown in Fig. 6. Figure 5: Well production data simulated rom the averaged permeability models updated by EnKF at dierent time.

11 SPE Real-Time Reservoir Model Updating Usign Ensemble Kalman Filter 11 Figure 9: Production data simulated rom 50 realizations updated at 300 days by EnKF using dierent ensemble sizes. Results rom the reerence model are in red. Figure 8: The mean/average and variance models at 120 days updated by EnKF using dierent ensemble sizes.

12 12 X.-H. Wen and W. H. Chen SPE Figure 10: The evolution o mean/average and variance ields rom 200 realizations at dierent assimilation time using isotropic covariance model. Figure 11: First row: the reerence model and its pressure and water saturation ield at 30 days. Second row: one initial permeability model and its pressure and water saturation ield at 30 days. Third row: updated permeability, pressure and water saturation ields by Kalman ilter. Bottom row: updated pressure and water saturation ield at 30 days by conirming to the updated permeability model.

13 SPE Real-Time Reservoir Model Updating Usign Ensemble Kalman Filter 13 Figure 13: One permeability model updated at 120 days by conirming EnKF (a) and no-conirming EnKF (b). The dierence between (a) and (b) is shown in (c). The predictions o BHP at P1 using (a) and (b) are shown in (d) with red being the result rom the reerence model. Figure 12: First row: the reerence model and its pressure and water saturation ield at 120 days. Second row: one initial permeability model (already assimilated data at 30, 60 and 90 days) and its pressure and water saturation ield at 120 days. Third row: updated permeability, pressure and water saturation ields by Kalman ilter. Bottom row: updated pressure and water saturation ield at 120 days by conirming to the updated permeability model.

14 14 X.-H. Wen and W. H. Chen SPE Figure 15: Production data at selected wells simulated by using updated realizations at 600 days when only assimilating the production data at the original 5 wells (P1-P4 and I). Results rom the reerence model are in red. Figure 14: The evolution o mean/average and variance rom 200 realizations at dierent assimilation times where new wells (P5- P8) are added at 120 days. Figure 16: Production data at selected wells simulated using updated realizations at 600 days when assimilating the production data at all 9 wells (P1-P8 and I). Results rom the reerence model are in red.