SPE-104255-PA-P - Streamline-Assisted Ensemble Kalman...

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Unformatted text preview: Streamline-Assisted Ensemble Kalman Filter for Rapid and Continuous Reservoir Model Updating Elkin Arroyo-Negrete, Deepak Devegowda, and Akhil Datta-Gupta, SPE, Texas A&M University, and J. Choe, SPE, Seoul National University Summary The use of the ensemble Kalman filter (EnKF) is a promising approach for data assimilation and assessment of uncertainties during reservoir characterization and performance forecasting. It provides a relatively straightforward approach to incorporating diverse data types, including production and/or time-lapse seismic data. Unlike traditional sensitivity-based history matching methods, the EnKF relies on a cross-covariance matrix computed from an ensemble of reservoir models to relate reservoir properties to production data. For practical field applications, we need to keep the ensemble size small for computational efficiency. However, this leads to poor approximations of the cross-covariance and, often, loss of geologic realism through parameter overshoots, in particular by introducing localized patches of low and high permeabilities. Because the EnKF estimates are “optimal” only for Gaussian variables and linear dynamics, these difficulties are compounded by the strong nonlinearity of the multiphase history matching problems and for non-Gaussian prior models. Specifically, the updated parameter distribution tends to become multiGaussian with loss of connectivities of extreme values, such as high permeability channels and low permeability barriers, which are of special significance during reservoir characterization. We propose a novel approach to overcome some of these limitations by conditioning the cross-covariance matrix using information gleaned from streamline trajectories. Our streamline-assisted EnKF is analogous to the conventional assisted history matching, whereby the streamline trajectories are used to identify gridblocks contributing to the production response of a specific well. We then use these gridblocks only to compute the cross-covariance matrix and eliminate the influence of unrelated or distant observations and spurious correlations. We show that the streamline-assisted EnKF is an efficient and robust approach for history matching and continuous reservoir model updating. We illustrate the power and utility of our approach using both synthetic and field applications. Introduction Proper characterization of the reservoir and the assessment of uncertainty are crucial aspects of any optimal reservoir development plan and management strategy. To achieve this goal, it is necessary to reconcile geological models to the dynamic response of the reservoir through history matching. The topic of history matching has been of great interest and an area of active research in the oil industry (Datta-Gupta and King 2007; Emanuel and Milliken 1998; Oliver et al. 2001). The past decade has seen some significant developments in assisted and automatic history matching of high-resolution reservoir models and associated uncertainty quantification. Many of these techniques involve computation of sensitivities that relate changes in production response at a well to a change in reservoir parameters. Techniques of automatic history matching that typically do not use parameter sensitivities or gradient of the misfit function are stochastic algorithms such as Copyright © 2008 Society of Petroleum Engineers This paper (SPE 104255) was accepted for presentation at the 2006 International Oil & Gas Conference and Exhibition in China, Beijing, 5–7 December, and revised for publication. Original manuscript received for review 21 August 2006. Revised manuscript received for review 8 July 2008. Paper peer approved 22 July 2008. 1046 Markov Chain Monte Carlo (MCMC), simulated annealing and genetic algorithms (Ma et al. 2008; Sen et al. 2005). A relatively recent and promising addition to this class of techniques is the use of ensemble Kalman Filters (EnKF) for data assimilation (Gu and Oliver 2005, 2006; Naevdal et al. 2005; Gao et al. 2006; Skjervheim et al. 2007; Dong et al. 2006). It is a Monte-Carlo approach that works with an ensemble of reservoir models. Specifically, the method utilizes cross-covariances between measurements and model parameters computed directly from the ensemble members to sequentially update the reservoir models. A major advantage of the EnKF is that it can be readily linked to any existing reservoir simulator. The ability to assimilate diverse data types and the ease of implementation have resulted in considerable interest in the approach. Moreover, EnKF uses a sequential updating technique; that is, the reservoir data is assimilated as and when it becomes available. The EnKF can assimilate the latest production data without re-running the simulator from the initial conditions. These characteristics make it particularly well-suited for continuous model updating. The increased application of downhole monitors, intelligent well systems, and permanent sensors to continuously record pressure, well rates, and temperature has provided a further boost to the sequential model updating through EnKF. In spite of all its favorable properties, the current implementation of EnKF approach comes with its own share of challenges. A key requirement in history matching is that the final model should honor the available geological information and retain geologic realism. It has been shown that the EnKF works well when the prior distribution of parameters is Gaussian; however, the estimates are suboptimal for non-Gaussian distributions. Over a sequence of many updates, multimodal permeability distributions tend to transform to Gaussian distribution. During geologic model updating, this can lead to a loss of structure and connectivity of the extremes in the permeability field. This has serious implications in the fluid flow because of the influence of high-permeability channels and low-permeability barriers. Although there are some variants of the Kalman filter that work with non-Gaussian distributions, such as the Gaussian summation approximation, the implementation on an ensemble framework tend to be very expensive (Anderson and Moore 1979). In the past few years, we have seen several applications of the EnKF for field-scale history matching, including some recent papers that attempt to deal with some of the challenges pertaining to its use (Gu and Oliver 2005, 2006; Naevdal et al. 2005; Gao et al. 2006; Skjervheim et al. 2007; Dong et al. 2006). In particular, localized overshooting of permeabilities has been reported, resulting in loss of geologic continuity. This is aggravated by the strong non-linearity inherent in multiphase flow simulations. Another common difficulty experienced when using the EnKF is filter divergence. The effect of filter divergence is such that the distribution produced by the filter drifts away from the truth. Filter divergence normally occurs because the prior probability distribution becomes too narrow (loss of variance) and the observations have progressively less impact on the model updates. One common approach to deal with filter divergence is to add some (white) noise to the prior ensemble to “inflate” its distribution and enhance the impact of new observations. Other problems and limitations of the EnKF, particularly for nonlinear problems December 2008 SPE Reservoir Evaluation & Engineering and non-Gaussian parameter distributions, can be partly controlled using a large ensemble. However, for practical field applications, the ensemble size needs to be kept relatively small for computational efficiency. This paper describes an approach to address many of the currently reported difficulties in the use of the EnKF applied to reservoir history matching. The unique feature of our proposed approach is that the final models that constitute the ensemble tend to retain the geological information that went into building them initially. Over a sequence of many updates, our approach tends to preserve the shape of the initial permeability distribution and consequently retains key geological features. Our approach greatly decreases the severity of the overshooting problem reported in earlier implementations of the EnKF. Moreover, it allows the use of smaller ensemble size, while providing results comparable or better than the standard EnKF. The paper is organized as follows. First, we briefly review the major steps of the EnKF and the additional streamline-based conditioning of the cross-covariance proposed here. We also illustrate these steps using a synthetic example. Next, we discuss the underlying mathematical formulation in detail. We then demonstrate the power and practical utility of the approach using the benchmark PUNQ-S3 synthetic example (Gu and Oliver 2005) and a field example. Finally, an analysis of the scalability and speed-up factor for the parallel implementation of our code is given. blocks that may be far away from the producers, particularly for highly heterogeneous reservoirs and in the presence of flow channels. The major steps in our proposed approach are outlined next. Approach A general rule for history matching is to change the parameters where the uncertainties are large and/or where the changes in the parameters will have the largest influence on the solution. Thus, it is vital to identify these regions and then limit the changes to these areas. Our knowledge of the reservoir drive mechanisms and underlying flow physics can be used to infer these regions. For instance, in primary depletion, the bottomhole pressure is mainly affected by reservoir parameters within the region defined by the radius of investigation. Similarly, during waterflooding, the watercut at the producing wells is affected primarily by the rock-fluid properties within the swept zone. A convenient way to identify these regions of influence during history matching is through examination of the streamline trajectories and the time of flight (Datta-Gupta and King 2007). The use of streamlines to decide regions for changes during reservoir history matching has proved to be useful in the past. Such streamline-assisted history matching was first proposed by Emanuel and Milliken (1998). They used streamlines to identify the gridblocks that affect the production response in a specific well. When these gridblocks are identified, it is possible to restrict changes to these gridblocks preferentially. Our proposed streamline-assisted EnKF shares many of the features discussed previously. Specifically, streamlines and time of flight are first used to identify areas of influence during history matching. The cross-covariance calculations that relate reservoir parameters to production data are then limited to these regions of influence. Obviously, these regions will change with time as the flood front progresses or the well conditions change. At each time, the streamlines are used to localize the cross-covariance calculations. Such conditioning of the cross-covariance has also been used in other applications such as weather forecasting, where the medium under consideration is more homogeneous (Houtekamer and Mitchell 2001; Hamill et al. 2001). For example, Houtekamer and Mitchell (2001) propose the use of an isotropic cutoff radius beyond which the influence of a given observation over the model parameters is considered to be small. The parameters outside the cutoff radius are then either damped or excluded during the computation of the cross-covariance matrix. By doing so, they avoided the estimation of small or erroneous correlation associated with remote locations. To filter remote observation, they used the Schur product of the covariance times a correlation function. A similar idea is used here, but we use streamlines to decide which gridblocks are strongly correlated to an observation. Our approach is more physically based and intuitive. The use of streamlines is also better-suited for reservoir problems compared to the cutoff-radius approach, because production responses are often related to grid- The Kalman Update. Update the reservoir model using the Kalman update equation (Eq. 4, discussed later). Repeat all steps again until all production data are assimilated. An outline of the procedure for our proposed approach is given in the flow chart in Fig. 1 and can be easily incorporated in any existing EnKF code. It only requires an additional step of streamline-based covariance localization compared to a standard EnKF implementation. December 2008 SPE Reservoir Evaluation & Engineering Ensemble Forecast Step. We start with an ensemble of reservoir models conditioned to static data. For each member of the ensemble, we simulate the production response up to the next available observation time, using either a streamline or a finitedifference simulator. For finite-difference models, we also trace the streamlines for each member of the ensemble. The streamline tracing is performed using the total phase fluxes from the simulator (Jimenez et al. 2007). Computation and Conditioning of the Cross-Covariance Matrix. For each member, we utilize the streamlines to associate gridblocks or regions that contribute to each producer at a given time. Next, we stack the selected gridblocks/regions from all ensemble members to define a common region of influence for all members. Using the production responses at current timestep from each member, we compute the cross-covariance matrix, including only the gridblocks within the common region of influence. Computation of the Kalman Gain. Using the covariance of the observed data, the model response and the cross-covariance from the previous step, compute the Kalman gain (Eq. 6 given later). An Illustration of the Procedure. The detailed mathematical formulation behind our proposed approach will be discussed later. First, we illustrate the procedure using a synthetic 2D example. For this purpose, we will refer to the approach proposed by Evensen (1994, 2003) and later introduced to reservoir history matching by Nævdal et al. (2005) as the “standard EnKF” and the streamlinebased conditioning approach proposed here as the “SL EnKF.” We set up a 2D reference model with a bimodal distribution of the permeability. A non-Gaussian distribution was chosen deliberately because of the difficulties encountered by standard EnKF for such distributions. The reservoir consists of a 2D area divided into 41×41 gridblocks. We simulate waterflood for 2,000 days with eight producers and one injector as shown in Fig. 2. Both fluids and the rock are assumed to be incompressible for this case. The GSLIB (1992) was used to generate the initial ensemble of 100 models satisfying the prior distribution and spatial continuity (Fig. 3a). Water cuts and bottomhole pressures from the reference model were treated as observed data for history matching using the EnKF. Results From the Standard EnKF. Fig. 3b shows the updated ensemble mean permeability and some ensemble members after the application of the standard EnKF. Notice that much of the geologic continuity in the initial model is absent from the updated models. This is because of the tendency of the standard EnKF to transform the log-permeability statistics to a Gaussian distribution after many updates. The log-permeability histogram for one of the updated ensemble members is shown in Fig. 4 (center), which clearly shows the lack of the bimodal character. The fact that the prior model statistics is not preserved is also indicated by the nonlinear behavior of the Q-Q plot. Multivariate model parameter distributions derived from the updated realizations can also be assessed for normality (Vasco et al. 1996). The scaled and squared deviations of each ensemble member from the ensemble mean, represented by the respective Mahalanobis distances, tend to follow a chi-square distribution, which indicates posterior parameter distributions that are multi-Gaussian (Devegowda 2008). This can 1047 Fig. 2—Nine-spot waterflooding example; figure shows the porosity of the reference model and the position of the producer and injectors. vicinity of the producer shows a stronger and more linear relationship compared to gridblock 1,523. In fact, there is very little impact of the permeability for gridblock 1,523 on the water-cut response as is evident from the scatter in the cross-plot. For most practical applications with modest ensemble size, we do not have enough samples to adequately estimate this cross-covariance, resulting in noisy or spurious estimates. Hamill and Whitaker (2001) have shown that for such gridblocks where the correlation coefficient is small, assimilation of information using the inaccurate values of the cross-covariance might have a detrimental outcome in terms of poor state/parameter estimates and potential filter divergence. We intend to avoid this situation using our proposed streamline-based covariance localization approach. Fig. 1—Streamline-assisted ensemble Kalman filter flow chart. have serious ramifications in geologic modeling as the multiGaussian posterior models are incapable of reproducing the connectivities of the extremes, specifically flow channels and barriers. It is interesting to note that in spite of the loss in the geologic continuity, the final members are able to closely match the production data as shown in Fig. 5. Overall, the EnKF greatly reduces the spread of the model responses around the observations. This also shows the inherent nonuniqueness in history matching and that not all models reproducing history can be considered valid for prediction purposes. Next, we repeat the same history matching, but this time we condition the cross-covariance matrix using the streamline path information. The Need for Covariance Localization. An important aspect of the EnKF is to quantify the relationship between the state variables using the appropriate cross-covariance measure. For our particular example, the cross-covariance matrix relates the water cut and the permeability at each gridblock. The computation of this cross covariance can be explained as follows. The EnKF uses an ensemble of realizations of the permeability. For each realization, flow simulation is carried out to calculate the water-cut response. Fig. 6 shows a cross plot of the different values of the permeability for two selected gridblocks (gridblock 172 and 1,523) from the different ensemble members vs. the computed water cut at a particular assimilation time. Clearly, the gridblock 172, which is in close 1048 Conditioning the Cross-Covariance and the SL EnKF. In practical applications of the EnKF, we need to keep the ensemble size reasonable to minimize the computation cost. To accomplish this, we condition the cross-covariance matrix by selecting those gridblocks for which the correlation with production response is expected to be significant. Fig. 7 shows all the gridblocks crossed by streamlines arriving at the producer P8 at a particular time. The results for two ensemble members, 15 and 73, are shown here. We have also shown the result after stacking all the ensemble members. The highlighted gridblocks in the stack indicate those gridblocks that have a streamline passing through them and arriving at producer P8 at least for one member of the ensemble. This stacked representation now defines our region of influence for the entire ensemble and only the gridblocks within this region are included in the cross-covariance calculations. Fig. 3 shows a comparison of the results from the standard and the SL EnKF. It is clear that while the standard EnKF produces over/undershooting of the permeability, the SL EnKF does not. Also, a visual examination of the permeability fields indicate that the geologic continuity is maintained. Fig. 4 shows a comparison of the histograms of the prior and the updated permeabilities for one of the ensemble members. Unlike the standard EnKF, the bimodal distribution is preserved here. This is further reinforced by the Q-Q plot, which now shows a linear behavior. Overall, the results indicate that the SL EnKF appears to outperform standard EnKF in terms of preserving the geologic realism during history matching. Fig. 5 shows the initial and final spread of the water cut at wells P1, P2, and P4 using the SL EnKF. The results show that our proposed approach is able to assimilate the production data without difficulty. Comparison With Distance-Dependent Localization. To illustrate the benefits of using the SL EnKF, we perform a comparison between SL EnKF and EnKF using distance-dependant covariance December 2008 SPE Reservoir Evaluation & Engineering Fig. 3—(a) Initial permeability map for randomly selected members in the ensemble; (b) updated permeabilty after application of the standard EnKF; and (c) updated permeability after application of the SL EnKF. localization (Houtekamer and Mitchell 2001). For distancedependant localization, we consider a localizing function centered at the well and monotonically decreasing from a value of one at the well location to a value of zero at some predefined cutoff radius (Gaspari and Cohn 1999). Outside of this cutoff radius, it is assumed that the model parameters have no influence on the performance of the well. The choice of a cut-off radius is quite subjective and is a potential weakness of the approach. We consider the synthetic example shown in Fig. 2, which is 410 feet (41×10 feet) on each side and use a cutoff radius of 100 feet for each well. Fig. 8b shows the final spread of the water-cut from wells P1, P2, and P4 in comparison with the initial model predictions in Fig. 8a. The poor match underscores the importance of the proper choice of Fig. 4—Comparison of the log-permeability statistics for member 23. Left: Histogram of the reference log-permeability values and the Q-Q plot comparing the true and initial histograms. Center: Histogram of the updated log-permeability values and the Q-Q plot comparing the initial and updated histograms after using the standard EnKF. Right: The Q-Q plot comparing the initial and updated histograms after using the SL EnKF and the corresponding histogram of the updated log-permeability values. December 2008 SPE Reservoir Evaluation & Engineering 1049 Fig. 5—Comparison of the water-cut spread for initial members (first row), the standard EnKF (second row), and the SL EnKF (third row). Fig. 6—Cross plot of ln permeability vs. water cut at Well P1 for different realizations at two gridblocks. Notice how the correlation coefficient is much higher for the gridblock closer to Well P1. 1050 December 2008 SPE Reservoir Evaluation & Engineering Fig. 7—Gridblocks (left and center) selected by the streamlines arriving producer P8 at time 750 days for members 15 and 73. Conditioning criteria for the covariance matrix for Well P8: the region of influence (right) is generated by stacking the gridblocks from all the members at time 750 days. localization. Next, we chose a larger cutoff radius of 250 feet. This choice seems to be more appropriate, as seen in Fig. 8c. However, the benefits of using the streamline-based localization are evident when we compare the permeability distribution for an arbitrarily chosen ensemble member shown in Fig. 9. Clearly, the streamline based covariance localization keeps the changes minimal and is able to better preserve the characteristics of the prior permeability field. In general, the selection of the cutoff radius for distancedependent localization will require trial and error. Furthermore, the localization may not be consistent with the underlying heterogeneity that might include high-permeability channels with contributions from large distances. Using streamlines, we can effectively define regions of influence with a sound physical basis that is tied to the underlying flow field and geology. Mathematical Background This section briefly reviews the standard EnKF and discusses the implementation of our proposed enhancements using streamlines. A more detailed description including the underlying theoretical Fig. 8—(a) Initial water-cut predictions. The reference model is shown in red. (b) Water-cut predictions from the updated members using a cutoff radius of 100 feet. The matches to the observed data in red are not satisfactory. (c) Water-cut predictions from the updated members using a cutoff radius of 250 feet. December 2008 SPE Reservoir Evaluation & Engineering 1051 Fig. 9—Updated permeability for one realization. (a) The initial model; (b) the updated model using distance-dependent localization, and (c) the updated model using the SL EnKF. basis and derivations of EnKF can be found in several earlier publications (Naevdal et al. 2005; Evensen 1994, 2003). Reservoir State Vector. In EnKF, the joint model parameter-data state vector y includes three types of elements: static variables ms k (e.g., permeabilities, porosity), dynamic variables obtained from flow simulation md (e.g., pressure, phase saturation), and the obk served production data dk (e.g., bottomhole pressure, well production rates, water cuts measured at the wells). Ά· ms k yp = k md k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) dk The state vector at time k is defined by Eq. 1, where superscript p denotes prior, s stands for static and, d stands for dynamic. One characteristic of the EnKF is that it uses an ensemble of state vectors to estimate the mean and the covariance. The ensemble of state vectors is represented by Eq. 2. p ⌿p = ͕yp yp . . . yk,Ne͖, . . . . . . . . . . . . . . . . . . . . . . . . . . (2) k k,1 k,2 where Ne represents the number of ensemble members. Each state vector represents an individual member of an ensemble of possible states that are consistent with the initial measurements from core, well logs, and seismic data. The static parameters of the ensemble members can be generated using appropriate geostatistical techniques such as sequential Gaussian simulation, indicator simulation, or by other means (Deutsch and Journel 1992). In general, each ensemble member will result in a different forecast and we do not know a priori which one is closer to the truth, if any. The best we can do is to provide an uncertainty analysis from the ensemble. A general way to describe a model state invokes a probability distribution over the model space. Our goal is to explore the model space by suitable stochastic simulation techniques to sample an ensemble of models that have enough “broadness” to provides a good estimation of the central tendency (mean) and dispersion (covariance matrix). The Forecast and the Update Steps. Ensemble Kalman filters have two main steps: a forecast step and an update step. In this work, the forecast step is carried out by a commercial streamline simulator (Frontsim 2005). This action can be represented in our equation notation as follows: ͭͮ md k dk = f ͑ms , md ͒, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) k−1 k−1 where f represents a numerical solution of the porous media fluid flow equations moving forward from time step k–1 to timestep k. The forecast step is followed by the update step, whereby the state variables are updated using the Kalman update equation as follows (Evensen 2003): 1052 ⌿u = ⌿p + ⌲͑Dk − H⌿p͒. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4) k k k The superscript u denotes updated and p denotes prior. Here, matrix K is the Kalman gain and the matrix D represents an ensemble of sampled observations, both defined later. The measurement matrix H is given below, where I is simply the identity matrix. H = ͓0 I͔. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) The basic function of the matrix H in Eq. 4 is to select rows in ⌿p k corresponding to the calculated production data dk. The Kalman gain matrix is given as follows (Evensen 2003): K = Cp HT͑HCp HT + CD͒−1, . . . . . . . . . . . . . . . . . . . . . . . . . . . (6) ⌿ ⌿ where Cp represent the state vector covariance matrix; and CD ⌿ represent observation covariance matrix. The ensemble of sampled observations Dk can be represented as follows: Dk = ͕dk,1 dk,2 . . . dk,Ne͖, . . . . . . . . . . . . . . . . . . . . . . . . . . (7) dk,i = dk + ␧i, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8) where dk represents a vector of any type of production data measured at time k, and ␧i represent the noise in the observation for each member i. The noise is assumed to be normally distributed with mean zero a covariance given by CD. Because the true state vector is not known, we approximate it with the mean of the ensemble. Then the covariance matrix Cp can ⌿ be computed at any point in time. yp = 1 Ne Cp = ⌿ Ne ͚y, p i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9) i=1 1 Ne − 1 Ne ͚ ͑y − y ͒͑y − y ͒ . p i p p j p T . . . . . . . . . . . . . . . . . . . . (10) i,j=1 It is important to point out that the Kalman update in Eq. 4 is a minimum variance estimate that is optimal only for Gaussian variables and linear dynamics. These conditions are typically violated for reservoir history matching, rendering the estimates suboptimal. Nevertheless, the results can be useful from practical point of view. In the Kalman gain equation the covariance appears always multiplied by the matrix H. Thus, in practice, there is no need to compute the whole covariance matrix as we require only a small portion of it. In our proposed approach here, we will further condition the matrix Cp to include only the regions that are crossed by ⌿ the data-relevant streamline trajectories. We call this streamlinebased covariance localization. To account for the conditioning using streamlines, the covariance matrix is redefined as Cp HT = ␳ ‫ؠ‬ ⌿ ͩ 1 Ne − 1 Ne ͚ ͑y − y ͒͑Hy − Hy ͒ p i i,j=1 p p j p T ͪ , . . . . . . . (11) December 2008 SPE Reservoir Evaluation & Engineering where ␳ is a correlation function (matrix) discussed next and represents the flow path information extracted from the streamlines (see Fig. 7). The operation ␳° in Eq. 11 denotes the Schur product operator, which is an element-by-element multiplication of the matrices (Houtekamer and Mitchell 2001; Hamill et al. 2001). We can think of the correlation function ␳ as a matrix with the column j filled with ones at the grid locations i selected in Fig. 7. For other gridblocks in the same column, the correlation function is set equal to zero. A similar procedure is repeated for all other producers j until matrix ␳ij is completed. We can build the correlation function at each assimilation time. In fact, it is possible to define different types of the correlation functions depending upon physical considerations. In this work, we have investigated two types of correlation functions: one based on streamline (denoted by SL) trajectories and the other based on the time of flight and the front location as given next. ␳ij‫ 0ס‬Gridblock i is crossed by a SL arriving at well j 1 Gridblock i is not crossed by any SL arriving at well j ␳ij‫ 0ס‬Gridblock i is crossed by SLs arriving at well j and the gridblock is behind the saturation front 1 Gridblock i is not crossed by any SL arriving at well j or the gridblock is ahead of the water saturation front. The saturation front along the streamline at any time t is given by the following relationship (Datta-Gupta and King 2007; Carter): dfw ␶ = , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12) dSw t where dfw/dSw is the derivative of the fractional flow curve with respect to water saturation at breakthrough and ␶ is the time of flight along the streamlines. Although Eq. 12 is derived under restrictive conditions (uniform saturation and injection conditions), our experience indicates that it can be applied more generally for the limited purpose of covariance localization. Generally, for cases involving water injection, there is no observable difference between the two types of covariance localization. The use of streamline-derived information can easily account for changing field conditions. Because streamlines simply reflect the underlying flow field, changing field conditions, for example infill drilling, do not pose any problems. The streamlines are retraced at each pressure update to account for changes in the well operating conditions. Consequently, in our approach the correlation matrix ␳ changes with flow dynamics and highlights the association between wells and the grid locations for the conditions existing in the reservoir at the time of assimilation. Application and Discussion We demonstrate the application and advantages of the SL EnKF using two examples. First, we use the PUNQ-S3 reservoir model for quantitatively examining the advantages of the SL EnKF. The PUNQ-S3 application demonstrates the generality of our approach in reservoirs with three-phase flow and highly compressible fluids (free gas). Next, we illustrate the practical feasibility of our proposed approach using a field example. The PUNQ-S3 Example. The PUNQ-S3 reservoir model was developed in the European Union by a group of companies and universities. Detailed description of the model can be found elsewhere (Floris et al. 2001; Barker et al. 2001; Carter). The PUNQS3 reservoir model consists of 19×28×5 gridlocks, of which 1,761 are active. The gridblock sizes are ⌬x=180 ft, ⌬y=180 ft, and ⌬z ranges from 1.3 to 8.8 ft. The reservoir has a small gas cap in the center of a dome shape structure. It has a fault to the east and south and a strong aquifer zone to the west and north (see Fig. 10). The field initially contains six production wells located around the gas/oil contact. Because of the strong aquifer influence, no injection wells are present. All six producing wells were produced as follows: an extended well testing during the first year, then a shut-in period lasting the following 3 years, and finally a 4-year production period. The well testing period consists of four time December 2008 SPE Reservoir Evaluation & Engineering Fig. 10—Top surface map showing well locations [from Floris et al. (2001)]. windows, each of which is 3 months long with constant flow rate. The oil production rate is fixed at 150 sm3/day within the 4-year production period. All wells have a 2-week shut-in each year to collect shut-in pressure. The reservoir properties and an input data file can be downloaded from reservoir project website (Carter). In our historymatching examples, we used a commercial streamline simulator (Frontsim). Because of the differences in the simulator, we did not use the 16.5 years of production data given in the PUNQ-S3 website. Instead, we regenerated our reference production data. The initial ensemble members were generated using the GSLIB. Values for porosity at well locations and anisotropy information used to generate the true model are given on the PUNQS3 website. The horizontal and vertical permeabilities were calculated using a deterministic relationship proposed elsewhere (Barker et al. 2001): log͑kh͒ = 9.02␸ + 0.77 kv = 0.31kh + 3.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13) We history matched the model using the standard EnKF and the SL EnKF. Production data was assimilated for a period of 8 years (2,936 days). Next, using the updated models, we performed flow simulation for all the ensemble members, starting from time zero and forecast the production response for the next 8.5 years. We chose to start from the beginning rather than latest assimilation time to examine the ability of the updated permeabilities to preserve matches at earlier times and also to avoid any potential material-balance problems arising from phase updating. As mentioned earlier, there were no injection wells for this example. The streamlines arriving at the producer at a given time are indicative of the drainage area associated with the well. Clearly, this drainage area will change with time. At early assimilation times, the streamlines cover regions around the well only as shown in Fig. 11a. At later times, the streamlines cover the whole reservoir model as shown in Fig. 11b. For water-cut-like measurements, it is possible to further condition the flow path information using the saturation front location derived from Eq. 12. Our goal here is to demonstrate the advantages of the SL EnKF compared to standard EnKF. For this purpose, we conduct the history matching using different ensemble sizes: 30, 60, and 100 members. Each case was run using the standard EnKF and the SL EnKF. Fig. 12 shows a comparison of the initial and updated bottomhole pressure, gas-to-oil ratio, and water cut for both the standard EnKF and the SL EnKF using 100 members. Both meth1053 where M is the number of gridblocks, ktrue is the reference permeability, and kmean is the mean permeability from the ensemble. The RMS for the permeability was computed for all the cases with different numbers of members in the ensemble. Fig. 14a shows the results for this calculation. Notice that the RMS error for the natural logarithm of the permeability is less for the SL EnKF compared to the standard EnKF, especially at later assimilation steps. This trend is the same for all the cases with different ensemble sizes tested here. This seems to indicate a lessened, or at least delayed, tendency for the filter to diverge in the presence of the streamline-based covariance localization. Fig. 14b shows a comparison of the RMS for different number of members in the ensemble. Again, the SL EnKF appears to show improved performance. The difference in the RMS error is particularly significant for small ensemble sizes, indicating the potential advantage of the SL EnKF for large-scale field applications. Fig. 11—Flow path information from streamlines. (a) Zones affected by the streamlines arriving at each producer at early time. (b) Zones selected by streamlines arriving at each producer at later time. ods are able to significantly decrease the spread between the observed and calculated production response as we increase the number of assimilation steps. However, the advantages of the SL EnKF becomes more evident when we analyze the error in the static variable viz. the permeability. To compute the variance in permeability, we use the reference permeability as the truth and each members from the ensemble as one sample of the random variable. The variance at each gridblock can now be computed as follows: var͑ln͑k͒͒ = 1 Ne − 1 Ne ͚ ͑ln͑k ͒ − ln͑k i 2 ref ͒͒ , . . . . . . . . . . . . . . . (14) i=1 where ln(k) is the natural logarithm of the permeability for the ensemble member i at a given gridblock, kref is the reference permeability for the corresponding gridblock, and Ne is the ensemble size. This operation is performed for every gridblock. Fig. 13 shows a comparison of the variance maps for the standard EnKF and the SL EnKF. These maps were computed using the updated permeability members at time 2,936 days. As expected, the variance is small around the well position. This is especially evident at Layers 4 and 3, where the wells are completed. A closer look of the variance maps from both the techniques reveals that overall the SL EnKF results in smaller variance compared to the standard EnKF. This is simply a reflection of the lack of overshoot/ undershoot of the permeability for the SL EnKF. Thus, we are able to better preserve the geologic continuity. Although we have shown the maps from the 60 members ensemble only, similar results were obtained with ensembles sizes of 30 and 100. Additionally, one can also compute the deviation of the ensemble members from the ensemble mean. This quantity is an indicator of the variability among the ensemble members. Our experience shows a faster loss of variability over a sequence of assimilation steps with the standard EnKF as compared to the SL EnKF. Consequently, with SL EnKF, there is a lesser tendency for the ensemble to ignore newer observations and filter divergence. Another possible way to check for errors in the estimated permeability is to compare the mean permeability from the ensemble with the reference permeability using the RMS measure, RMSln͑k͒ = 1054 ͱ 1 M M ͚ ͑ln͑k true,i ͒ i=1 − ln͑kmean,i͒͒2, . . . . . . . . . . . (15) The Goldsmith Field Case. We now discuss the application of the standard EnKF to a field case and compare the results with the SL EnKF. The field case is from the Goldsmith San Andres Unit (GSAU), a dolomite formation in west Texas. We matched 20 years of waterflood production history. The pilot area (see Fig. 15) consists of nine inverted five-spot patterns covering approximately 320 acres with an average thickness of 100 ft. The area has more than 50 years of production history before the initiation of the CO2 project in 1996. Because of practical difficulties describing the correct boundary conditions for the pilot area, wells around the pilot area were included in this study. The study area includes 11 injectors and 31 producers. Production history information from only nine producers is used because only these have significant water-cut response. The detailed production rate and the well schedule, including infill drilling, well conversions, and well shutin, can be found elsewhere (He and Datta-Gupta 2001). The study area was discretized into 58×53×10 gridblocks. The initial 100 realizations of porosity and permeability were obtained using sequential Gaussian co-simulation conditioned to well and seismic data. Fig. 16 shows some of the initial members; notice the PDF is not Gaussian because of the presence of multiple geologic facies. Fig. 17 shows some of the ensemble members after assimilation of water-cut data. Notice that the permeability distribution after a sequence of assimilation steps became totally Gaussian. The standard EnKF was unable to preserve the initial density function of the permeability. The multi-Gaussian distribution leads to loss of flow channels and barriers because of its maximum entropy character. Also, initially the ensemble members have permeabilities ranging from 0.005 md to 500 md. However, after the data assimilation, the updated permeabilities range between 2×10−6 md and 1×106 md, which clearly indicates very large and unrealistic changes. Such overshooting/undershooting problems have also been observed by others (Gu and Oliver 2005, 2006; Naevdal et al. 2005). Fig. 18 shows the initial and final spread of water cut at the wells. As before, a significant reduction in spread is observed. Fig. 19 shows the ensemble mean permeability before and after updating. The loss of structure in the permeability field is quite apparent here. For this field example, the wells that contain watercut information are located toward the center of the model. Injectors are located around the producers (see Fig. 15). This suggests that most of the water arriving at producers probably travels across the reservoir volume surrounded by the injectors and the producing wells. Thus, the major changes from the standard EnKF should be preferentially found in the area subscribed by the injectors and wells with observations. Unfortunately, this is not the case here. The standard EnKF has produced changes in all parts of the model regardless of the wells and injector position (see Fig. 19). We now repeat the same history matching, but this time by conditioning the covariance matrix using the streamline path information. To select the gridblocks for a specific producer, we use only those streamlines that have broken through at a given time (see Eq. 12). Fig. 20 illustrates such regions, considering all the streamlines arriving at producer P-6 at three different times. The water-cut matches are shown in Fig. 18 and exhibit similar behavior as in the case of the standard EnKF. December 2008 SPE Reservoir Evaluation & Engineering Fig. 12—(a) BHP spread at Well PRO-15; (b) GOR at Well PRO-1; and (c) water cut at producer PRO-11. The red bold line is the reference. The gray line around 3,000 days shows time up to which the information was assimilated. December 2008 SPE Reservoir Evaluation & Engineering 1055 Fig. 13—Variance map of the natural logarithm of permeability at 2,936 days. (a) 60 members ensemble using the streamlineassisted EnKF; (b) 60 members ensemble using the standard EnKF. Fig. 14—RMS error from the mean of the ensemble. (a) RMS error for the natural logarithm of the permeability after each assimilation step. (b) RMS error for the natural logarithm of the permeability for the standard EnkF and the streamline-assisted EnKF for different number of members in the ensemble. The real advantage of using the SL EnKF becomes evident during analysis of the final permeability fields. Fig. 21 shows the updated permeability for some of the members in the ensemble. Notice how the SL EnKF tends to preserve the initial permeability distribution. Both the spatial continuity and the bimodal nature of the permeability distribution are maintained. Furthermore, the strong overshooting problems encountered while using the standard EnKF does not seem to exist here. An analysis of the changes proposed by the SL EnKF (see Fig. 19) reveals that the changes are small and mostly localized in water-swept zones, which is physically reasonable. Thus, the updated ensemble members are able to preserve and inherit the non-Gaussian distribution by limiting and localizing the changes to regions supported by the dynamic data. Parallel Implementation of the Algorithm and Scalability A major advantage of the EnKF is that parallelization of the code can be achieved with relatively little effort. Several authors have 1056 Fig. 15—Goldsmith study area; well distribution producer with water-cut information highlighted with circles; injectors are darker uncircled dots. [from He and Datta-Gupta (2001)]. December 2008 SPE Reservoir Evaluation & Engineering Fig. 16—Permeability fields for the Goldsmith case (randomly selected members: 17 left, 58 center, and 97 right.). Permeability maps generated using sequential Gaussian cosimulation conditioned to wells and seismic data. Below each map, the histogram of the permeability is shown. Fig. 17—Updated permeability maps using the standard EnKF conditioned to water cut. Below each map is shown the histogram. December 2008 SPE Reservoir Evaluation & Engineering 1057 Fig. 18—Water cut 100 members ensemble spread at Wells P1, P3, P7. Initial (top row), after standard EnKF (center row), and after SL EnKF (bottom row). The gray line around 4,000 days shows the time up to which the information was assimilated. Fig. 19—(First) Initial mean from the 100 members; (second) mean from the 100 members after standard EnKF update; (third) mean from the 100 members after SL EnKF update; (fourth) standard EnKF changes in permeability; and (fifth) SL EnKF changes in permeability. Fig. 20—Regions selected by the streamlines arriving to producer P6 (black circle) at different times: (left) time 1,680 days, (center) 2,280 days, and (right) 3,960 days. 1058 December 2008 SPE Reservoir Evaluation & Engineering Fig. 21—Updated permeability map using the SL EnKF after water cut assimilation; below each map is shown the corresponding histograms. Notice the SL EnKF is able to preserve the bimodal prior density function. recognized the feasibility of parallel computation for the EnKF analysis. A rather simplistic approach to parallelize any EnKF implementation will require running each forward model in parallel. Further parallelization can be done by parallelizing the Kalman update equation as discussed by other authors (Keppenne 2000). In our parallel implementation, flow simulation for each member is performed in a different CPU (node). Results from each member are written by the simulator in binaries to the same network file system (NFS). After each forward run, information is retrieved from the NFS by the master node with a minimal level of interprocessor communication. Most of the examples presented in this paper could be run within acceptable CPU time. Synthetic cases with grid sizes of 50×50×1 and 100 ensemble members typically took from 2 to 3 hours to assimilate 25 to 40 observations in a single CPU computer. While running our field case example on a single CPU computer, it would take from 20 to 25 hours to assimilate the observation data for an ensemble of 100 members. We used the Message Parsing Interface (MPI) to parallelize our code. Tests were run in the 128 CPU shared-memory supercomputer at Texas A&M University. Even from our rather unoptimized parallel implementation, the results appear to indicate that the EnKF scales quite well in terms of CPU time as shown in Fig. 22. It is clear from our experiments that the application of the SL EnKF for continuous data assimilation into high-resolution reservoir models is quite feasible with the current level of computing power. Conclusions We have presented a novel streamline-assisted EnKF for continuous model updating using the selective flow path information from streamlines. The approach avoids much of the problems associated with the standard EnKF related to parameter overshoots and loss of geologic realism during history matching. We demonstrate the power and utility of our proposed approach using both synthetic and field examples. Some specific conclusions from this paper are summarized next. The streamline trajectory-based covariance localization appears to eliminate and/or minimize previously reported problems when using the standard EnKF for reservoir history matching. Some of the reported problems include overshooting of the reservoir parameters, the loss of variance, and the loss of geologic continuities for nonlinear problems or non-Gaussian distribution, particularly for modest ensemble sizes (<100). When comparing the standard EnKF and the streamlineassisted EnKF in terms of reservoir static parameters (for example, permeability), the SL EnKF is able to better preserve the histograms and the spatial continuity. The standard EnKF tends to make the updated parameters Gaussian and results in loss of connectivities in the extreme values. This adversely impacts the performance forecasting because the flow channels and barriers are no longer represented properly in the geologic model. When comparing the standard EnKF and the SL EnKF in terms of their ability to match and reproduce historical production data, both approaches appear to perform equally well. The results presented here suggest that the streamline-based covariance localization may significantly improve the quality of results from the standard EnKF or its variants. The cost of localizing the covariance will be significantly less compared to the cost of generating a large enough ensemble to achieve the same level of accuracy. The parallel implementation of our approach appears to scale favorably with respect to model size and the number of members in the ensemble, making the approach suitable for history matching and uncertainty quantification using detailed geologic models. Nomenclature CD ‫ ס‬data observation covariance matrix Cp ‫ ס‬joint model parameter-data state vector covariance ⌿ matrix Cp Ht ‫ ס‬model data cross-covariance matrix ⌿ dfw dSw ‫ ס‬fractional flow derivative at the saturation front dk ‫ ס‬observed or calculated data Dk ‫ ס‬ensemble observation at time k H ‫ ס‬measurement matrix HCp Ht ‫ ס‬calculated data covariance matrix ⌿ kv /kh ‫ ס‬vertical/horizontal permeability K ‫ ס‬Kalman gain ms ‫ ס‬static variable vector at time k k md ‫ ס‬dynamic variable vector at time k k Ne ‫ ס‬number of members in the ensemble Fig. 22—(a) Total running time for the Goldsmith field case as a function of the number of processors. (b) Speedup factor for the same dataset. December 2008 SPE Reservoir Evaluation & Engineering 1059 yp ‫ ס‬prior joint model parameter-data state vector, at time k k y p ‫ ס‬mean of the prior joint mode parameter data state vector ␦ij ‫ ס‬Kroneker delta ␧i ‫ ס‬white random noise data observation ␳ ‫ ס‬streamline base correlation function ␶ ‫ ס‬streamline time of flight ␸ ‫ ס‬porosity ⌿k ‫ ס‬ensemble of joint model-data state vectors Acknowledgments The authors would like to acknowledge financial support from members of the Texas A&M Joint Industry Project, MCERI (Model Calibration and Efficient Reservoir Imaging) and from the U.S. Department of Energy. We also would like to thank the Texas A&M University Supercomputing Facility for their support. References Anderson, B.D. and Moore, J.B. 1979. Optimal Filtering. Englewood Cliffs, New Jersey: Prentice-Hall. Barker, J.W., Cuypers, M., and Holden, L. 2001. Quantifying Uncertainty in Production Forecasts: Another Look at the PUNQ-S3 Problem. SPEJ 6 (4): 433–441. SPE-74707-PA. 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Oliver, D.S., Reynolds, A.C., Bi, Z., and Abacioglu, Y. 2001. Integration of production data into reservoir models. Petroleum Geoscience 7 (Supplement, 1 May): 65–73. Sen, M.K., Datta-Gupta, A., Stoffa, P.L., Lake, L.W., and Pope, G.A. 1995. Stochastic Reservoir Modeling Using Simulated Annealing and Genetic Algorithms. SPEFE 10 (1): 49–55. SPE-24754-PA. DOI: 10.2118/24754-PA. Skjervheim, J.-A., Evensen, G., Aanonsen, S.I., and Johansen, T.A. 2007. Incorporating 4D Seismic Data in Reservoir Simulation Model Using Ensemble Kalman Filter. SPEJ 12 (3): 282–292. SPE-95789-PA. DOI: 10.2118/95789-PA. Vasco, D.W., Peterson, J.E. and Majer, E.L. 1996. Non-uniqueness in Travel-Time Tomography: Ensemble Inference and Cluster Analysis. Geophysics 61(4):1209-1227. DOI:10.1190/1.1444040. Elkin Arroyo is a reservoir engineer currently working for Occidental Oil & Gas. He has previously worked for the Colombian National Oil Company Ecopetrol ICP and for Numerica Limited, an engineering consulting firm in Colombia. He won first place in the SPE Golf Coast Region Student Paper Contest (2006). He holds a BS degree in mechanical engineering from the Universidad Industrial de Santander in Colombia and an MS degree in petroleum engineering from Texas A&M University. Deepak Devegowda is Assistant Professor in the Petroleum Engineering Department at the University of Oklahoma. His research is focused on problems related to reservoir characterization, geostatistics and unconventional oil, and gas recovery. He holds PhD and Masters degrees in petroleum engineering from Texas A&M University. Akhil Datta-Gupta is Professor and holder of the LeSuer Chair in the Petroleum Engineering Department at Texas A&M University. He holds a PhD degree in petroleum engineering from the University of Texas at Austin and has worked for BP Exploration/Research and the Lawrence Berkeley National Laboratory. He is the recipient of the 2003 SPE Lester C. Uren Award for significant technical contributions in petroleum reservoir characterization and streamline-based flow simulation. He is an SPE Distinguished Member (2001), Distinguished Lecturer (1999–2000), and Distinguished Author (2000) and was selected as an Outstanding Technical Editor in 1996. He also has received the Cedric K. Ferguson Certificate (2000, 2006) and the AIME Rossitter W. Raymond Award (1992). Jonggeun Choe is an associate professor in the department of energy system engineering at Seoul National University e-mail: [email protected] Choe is also a member of the Research Institute Of Engineering Science in the university. His main areas of research are well control and inverse modeling for data integration and characterization. Before joining the university, he was a research engineer at Texas A&M University and developed conventional and SMD wellcontrol simulators. He was the recipient of the AIME Rossiter W. Raymond Memorial Award in 2000. He holds BS and MS degrees in mineral and petroleum engineering from Seoul National University and a PhD degree in petroleum engineering from Texas A&M University. He is one of technical editors of SPE Drilling & Completion and a member of SPE, IAMG, and KSGE. December 2008 SPE Reservoir Evaluation & Engineering ...
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