Multiscale ensemble filtering for reservoir engineering applications

Reservoir management requires periodic updates of the simulation models using the production data available over time. Traditionally, validation of reservoir models with production data is done using a history matching process. Uncertainties in the data, as well as in the model, lead to a nonunique history matching inverse problem. It has been shown that the ensemble Kalman filter (EnKF) is an adequate method for predicting the dynamics of the reservoir. The EnKF is a sequential Monte-Carlo approach that uses an ensemble of reservoir models. For realistic, large-scale applications, the ensemble size needs to be kept small due to computational inefficiency. Consequently, the error space is not well covered (poor cross-correlation matrix approximations) and the updated parameter field becomes scattered and loses important geological features (for example, the contact between high- and low-permeability values). The prior geological knowledge present in the initial time is not found anymore in the final updated parameter. We propose a new approach to overcome some of the EnKF limitations. This paper shows the specifications and results of the ensemble multiscale filter (EnMSF) for automatic history matching. EnMSF replaces, at each update time, the prior sample covariance with a multiscale tree. The global dependence is preserved via the parent–child relation in the tree (nodes at the adjacent scales). After constructing the tree, the Kalman update is performed. The properties of the EnMSF are presented here with a 2D, two-phase (oil and water) small twin experiment, and the results are compared to the EnKF. The advantages of using EnMSF are localization in space and scale, adaptability to prior information, and efficiency in case many measurements are available. These advantages make the EnMSF a practical tool for many data assimilation problems.     

An enhanced ensemble Kalman filter scheme incorporating model error in sequential coupling between flow and geomechanics

In this work, we construct a new methodology for enhancing the predictive accuracy of sequential methods for coupling flow and geomechanics while preserving low computational cost. The new computational approach is developed within the framework of the fixed-stress split algorithm procedure in conjunction with data assimilation based on the ensemble Kalman filter (EnKF). In this context, we identify the high-fidelity model with the two-way formulation where additional source term appears in the flow equation containing the time derivative of total mean stress. The iterative scheme is then interlaced with data assimilation steps, which also incorporate the modeling error inherent to the EnKF framework. Such a procedure gives rise to an “enhanced one-way formulation,” exhibiting substantial improvement in accuracy compared with the classical one-way method. The governing equations are discretized by mixed finite elements, and numerical simulation of a 2D slab problem between injection and production wells illustrate the tremendous achievement of the method proposed herein.     

The temperature-based localization for the application of EnKF on automatic history matching of the SAGD process

The ensemble Kalman filter (EnKF) has been successfully applied to data assimilation in steam-assisted gravity drainage (SAGD) process, but applications of localization for the EnKF in the SAGD process have not been studied. Distance-based localization has been reported to be very efficient for assimilation of large amounts of independent data with a small ensemble in water flooding process, but it is not applicable to the SAGD process, since in the SAGD process, oil is produced mainly from the transition zone steam chamber to cold oil instead of the regions around the producer. As the oil production rate is mainly affected by the temperature distribution in the transition zone, temperature-based localization was proposed for automatic history matching of the SAGD process. The regions of the localization function were determined through sensitivity analysis by using a large ensemble with 1000 members. The sensitivity analysis indicated that the regions of cross-correlations between oil production and state variables are much wider than the correlations between production data and model variables. To choose localization regions that are large enough to include the true regions of non-zero cross-covariance, the localization function is defined based on the regions of non-zero covariances of production data to state variables. The non-zero covariances between production data and state variables are distributed in accordance with the steam chamber. This makes the definition of a universal localization function for different state variables easier. Based on the cross-correlation analysis, the temperature range in which oil production is contributed is determined, and beyond or below this range, the localization function reduces from one, and at the critical temperature or steam temperature, the localization function reduces to zero. The temperature-based localization function was obtained through modifying the distance-based localization function. Localization is applied to covariance of data with permeability, saturation, and temperature, as well as the covariance of data with data. A small ensemble (10 ensemble members) was employed in several case studies. Without localization, the variability in the ensemble collapsed very quickly and lost the ability to assimilate later data. The mean variance of model variables dropped dramatically by 95 %, and there was almost no variability in ensemble forecasts, while the prediction was far from the reference with data mismatch keeping up at a high level. At least 50 ensemble members are needed to keep the qualities of matches and forecasts, which significantly increases the computation time. The EnKF with temperature-based localization is able to avoid the collapse of ensemble variability with a small ensemble (10 members), which saves the computation time and gives better history match and prediction results.     

Comparing the adaptive Gaussian mixture filter with the ensemble Kalman filter on synthetic reservoir models

Over the last years, the ensemble Kalman filter (EnKF) has become a very popular tool for history matching petroleum reservoirs. EnKF is an alternative to more traditional history matching techniques as it is computationally fast and easy to implement. Instead of seeking one best model estimate, EnKF is a Monte Carlo method that represents the solution with an ensemble of state vectors. Lately, several ensemble-based methods have been proposed to improve upon the solution produced by EnKF. In this paper, we compare EnKF with one of the most recently proposed methods, the adaptive Gaussian mixture filter (AGM), on a 2D synthetic reservoir and the Punq-S3 test case. AGM was introduced to loosen up the requirement of a Gaussian prior distribution as implicitly formulated in EnKF. By combining ideas from particle filters with EnKF, AGM extends the low-rank kernel particle Kalman filter. The simulation study shows that while both methods match the historical data well, AGM is better at preserving the geostatistics of the prior distribution. Further, AGM also produces estimated fields that have a higher empirical correlation with the reference field than the corresponding fields obtained with EnKF.     

Application of the EnKF and Localization to Automatic History Matching of Facies Distribution and Production Data

The performance of the ensemble Kalman filter (EnKF) for continuous updating of facies location and boundaries in a reservoir model based on production and facies data for a 3D synthetic problem is presented. The occurrence of the different facies types is treated as a random process and the initial distribution was obtained by truncating a bi-Gaussian random field. Because facies data are highly non-Gaussian, re-parameterization was necessary in order to use the EnKF algorithm for data assimilation; two Gaussian random fields are updated in lieu of the static facies parameters. The problem of history matching applied to facies is difficult due to (1) constraints to facies observations at wells are occasionally violated when productions data are assimilated; (2) excessive reduction of variance seems to be a bigger problem with facies than with Gaussian random permeability and porosity fields; and (3) the relationship between facies variables and data is so highly non-linear that the final facies field does not always honor early production data well. Consequently three issues are investigated in this work. Is it possible to iteratively enforce facies constraints when updates due to production data have caused them to be violated? Can localization of adjustments be used for facies to prevent collapse of the variance during the data-assimilation period? Is a forecast from the final state better than a forecast from time zero using the final parameter fields?To investigate these issues, a 3D reservoir simulation model is coupled with the EnKF technique for data assimilation. One approach to enforcing the facies constraint is continuous iteration on all available data, which may lead to inconsistent model states, incorrect weighting of the production data and incorrect adjustment of the state vector. A sequential EnKF where the dynamic and static data are assimilated sequentially is presented and this approach seems to have solved the highlighted problems above. When the ensemble size is small compared to the number of independent data, the localized adjustment of the state vector is a very important technique that may be used to mitigate loss of rank in the ensemble. Implementing a distance-based localization of the facies adjustment appears to mitigate the problem of variance deficiency in the ensembles by ensuring that sufficient variability in the ensemble is maintained throughout the data assimilation period. Finally, when data are assimilated without localization, the prediction results appear to be independent of the starting point. When localization is applied, it is better to predict from the start using the final parameter field rather than continue from the final state.     

History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations

The ensemble Kalman filter (EnKF) has become a popular method for history matching production and seismic data in petroleum reservoir models. However, it is known that EnKF may fail to give acceptable data matches especially for highly nonlinear problems. In this paper, we introduce a procedure to improve EnKF data matches based on assimilating the same data multiple times with the covariance matrix of the measurement errors multiplied by the number of data assimilations. We prove the equivalence between single and multiple data assimilations for the linear-Gaussian case and present computational evidence that multiple data assimilations can improve EnKF estimates for the nonlinear case. The proposed procedure was tested by assimilating time-lapse seismic data in two synthetic reservoir problems, and the results show significant improvements compared to the standard EnKF. In addition, we review the inversion schemes used in the EnKF analysis and present a rescaling procedure to avoid loss of information during the truncation of small singular values.     

Reservoir Description with Integrated Multiwell Data Using Two-Dimensional Wavelets

Inferring reservoir data from dynamic production data has long been done through matching the production history. However, proper integration of available production history has always been a challenge. Different production history data such as well pressure and water cut often occur at different scales making their joint inversion difficult. Furthermore, production data obtained from the same well or even the same reservoir are often correlated making a significant portion of the dataset redundant. Thirdly, the massiveness of the data recorded from wells in a large reservoir over a long period of time makes the nonlinear inversion of such data computational demanding. In this paper, we propose the integration of multiwell production data using wavelet transform. The method involves the use of a two-dimensional wavelet transformation of the data space in order to integrate multiple production data and reduce the correlation between multiwell data. Multiple datasets from different wells, representing different production responses (pressure, water cut, etc.), were treated as a single matrix of data rather than separate vectors that assume no correlation amongst datasets. This enabled us to transform the multiwell production data into a two-dimensional wavelet domain and subsequently select the most important wavelets for history match. By minimizing the square of the Frobenius norm of the residual matrix we were able to match the calculated response to the observed response. We derived the relationship that allows us to replace a conventional minimization of the sum of squares of the l ₂ norms of multi-objective functions with the minimization of the square of the Frobenius norm of the integrated data. The usefulness of the approach is demonstrated using two examples. The approach proved very effective at reducing correlation between multiwell data. In addition, the method helped to reduce the cost of computing sensitivity coefficients. However, the method gave poor prediction of water cut when the datasets were not scaled before inverse modeling.     

Pattern Recognition in a Bimodal Aquifer Using the Normal-Score Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) is now widely used in diverse disciplines to estimate model parameters and update model states by integrating observed data. The EnKF is known to perform optimally only for multi-Gaussian distributed states and parameters. A new approach, the normal-score EnKF (NS-EnKF), has been recently proposed to handle complex aquifers with non-Gaussian distributed parameters. In this work, we aim at investigating the capacity of the NS-EnKF to identify patterns in the spatial distribution of the model parameters (hydraulic conductivities) by assimilating dynamic observations in the absence of direct measurements of the parameters themselves. In some situations, hydraulic conductivity measurements (hard data) may not be available, which requires the estimation of conductivities from indirect observations, such as piezometric heads. We show how the NS-EnKF is capable of retrieving the bimodal nature of a synthetic aquifer solely from piezometric head data. By comparison with a more standard implementation of the EnKF, the NS-EnKF gives better results with regard to histogram preservation, uncertainty assessment, and transport predictions.     

基于局域化EnKF同时同化地下水水流和溶质运移数据的渗透系数场识别研究

局域化改进集合卡尔曼滤波（EnKF）可以克服EnKF方法在使用小集合时,对参数识别精度较低的缺陷,其能同化 地下水位观测数据有效识别渗透系数场。实际工作中,溶质运移数据也较容易获得。崔凯鹏（2013）尝试增加溶质运移 数据以改进只同化水流数据对渗透系数的估计结果,但是精度提高有限。本文在其基础上修改模型,进一步增加溶质注 入井,探究同时同化水头和溶质运移数据,对渗透系数场识别效果,之后对比了局域化EnKF与非局域化EnKF参数识别结 果,并分析了溶质影响范围与参数识别的关系。结果表明：同时同化溶质运移和水头资料,比同化单一种类观测数据识别 的渗透系数精度更高;相同实现数目下,局域化EnKF比EnKF对渗透系数场的估计结果与真实场更为接近;仅考虑溶质影 响范围内的渗透系数,同化水头数据在最后时刻参数识别结果好于同化溶质运移数据参数识别结果,但差别不大。     

Continuous Facies Updating Using the Ensemble Kalman Filter and the Level Set Method

We present a methodology based on the ensemble Kalman filter (EnKF) and the level set method for the continuous model updating of geological facies with respect to production data. Geological facies are modeled using an implicit surface representation and conditioned to production data using the ensemble Kalman filter. The methodology is based on Gaussian random fields used to deform the facies boundaries. The Gaussian random fields are used as the model parameter vector to be updated sequentially within the EnKF when new measurements are available. We show the successful application of the methodology to two synthetic reservoir models.     

Ensemble Kalman filtering for non-linear likelihood models using kernel-shrinkage regression techniques

One of the major limitations of the classical ensemble Kalman filter (EnKF) is the assumption of a linear relationship between the state vector and the observed data. Thus, the classical EnKF algorithm can suffer from poor performance when considering highly non-linear and non-Gaussian likelihood models. In this paper, we have formulated the EnKF based on kernel-shrinkage regression techniques. This approach makes it possible to handle highly non-linear likelihood models efficiently. Moreover, a solution to the pre-image problem, essential in previously suggested EnKF schemes based on kernel methods, is not required. Testing the suggested procedure on a simple, illustrative problem with a non-linear likelihood model, we were able to obtain good results when the classical EnKF failed.     

Applying an optimized proxy-based workflow for fast history matching

History matching is still one of the main challenging parts of reservoir study especially in giant brown oil fields with lots of wells. In these cases, history matching with conventional manual technique needs many runs and takes months to get a match. In this work, an innovative approach was suggested for fast history matching in a real brown field. The workflow was employed based on an optimized proxy model for history matching of a field consisting of 14 active wells with multiple responses (which are production rate and pressure data) in the south part of Iran. The main important features of the proposed algorithm were defining a proxy model which is response surface method in which 21 model parameters were incorporated based on cubic centered face method. The proxy model was then optimized by one of the most famous algorithms which is genetic algorithm. Proxy model was successfully performed using 256 samples leading into p- value of 0.531 and R ² of 0.91 dataset. As a result, the proposed workflow and algorithm showed good and acceptable results for history matching of studied real model.     

Conditioned Simulations of Random Velocity Fields

In petroleum exploration and production, it is essential to have good estimations of the uncertainties on the reserves. Uncertainties on the velocity model used during the data processing are of major importance in this estimation. The generation of several velocity fields gives access to a quantified estimation of the uncertainties due to the velocity model inversion. The use of statistical methods helps in generating several important, equiprobable velocity fields, matching all the available velocity information. This paper presents an efficient simulation algorithm to generate instantaneous velocity fields, constrained by the distribution of values measured at the wells, and calibrated by the stacking velocities, taken as root-mean-square velocities. The simulations also match the covariance model given for the instantaneous velocity fields. The method is developed in a simple one-layer case with constant velocity, and then extended to more realistic situations. Finally, a real data application is shown, using data provided by ENI–Agip Division, and the efficiency of the proposed simulation method is discussed.     

Determination of lower and upper bounds of predicted production from history-matched models

We present a method to determine lower and upper bounds to the predicted production or any other economic objective from history-matched reservoir models. The method consists of two steps: 1) performing a traditional computer-assisted history match of a reservoir model with the objective to minimize the mismatch between predicted and observed production data through adjusting the grid block permeability values of the model. 2) performing two optimization exercises to minimize and maximize an economic objective over the remaining field life, for a fixed production strategy, by manipulating the same grid block permeabilities, however without significantly changing the mismatch obtained under step 1. This is accomplished through a hierarchical optimization procedure that limits the solution space of a secondary optimization problem to the (approximate) null space of the primary optimization problem. We applied this procedure to two different reservoir models. We performed a history match based on synthetic data, starting from a uniform prior and using a gradient-based minimization procedure. After history matching, minimization and maximization of the net present value (NPV), using a fixed control strategy, were executed as secondary optimization problems by changing the model parameters while staying close to the null space of the primary optimization problem. In other words, we optimized the secondary objective functions, while requiring that optimality of the primary objective (a good history match) was preserved. This method therefore provides a way to quantify the economic consequences of the well-known problem that history matching is a strongly ill-posed problem. We also investigated how this method can be used as a means to assess the cost-effectiveness of acquiring different data types to reduce the uncertainty in the expected NPV.     

Ensemble filter methods with perturbed observations applied to nonlinear problems

The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g., water cut and gas–oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble methods when applied to nonlinear problems.     

Direct forecasting of reservoir performance using production data without history matching

The conventional paradigm for predicting future reservoir performance from existing production data involves the construction of reservoir models that match the historical data through iterative history matching. This is generally an expensive and difficult task and often results in models that do not accurately assess the uncertainty of the forecast. We propose an alternative re-formulation of the problem, in which the role of the reservoir model is reconsidered. Instead of using the model to match the historical production, and then forecasting, the model is used in combination with Monte Carlo sampling to establish a statistical relationship between the historical and forecast variables. The estimated relationship is then used in conjunction with the actual production data to produce a statistical forecast. This allows quantifying posterior uncertainty on the forecast variable without explicit inversion or history matching. The main rationale behind this is that the reservoir model is highly complex and even so, still remains a simplified representation of the actual subsurface. As statistical relationships can generally only be constructed in low dimensions, compression and dimension reduction of the reservoir models themselves would result in further oversimplification. Conversely, production data and forecast variables are time series data, which are simpler and much more applicable for dimension reduction techniques. We present a dimension reduction approach based on functional data analysis (FDA), and mixed principal component analysis (mixed PCA), followed by canonical correlation analysis (CCA) to maximize the linear correlation between the forecast and production variables. Using these transformed variables, it is then possible to apply linear Gaussian regression and estimate the statistical relationship between the forecast and historical variables. This relationship is used in combination with the actual observed historical data to estimate the posterior distribution of the forecast variable. Sampling from this posterior and reconstructing the corresponding forecast time series, allows assessing uncertainty on the forecast. This workflow will be demonstrated on a case based on a Libyan reservoir and compared with traditional history matching.     

Integration of Markov mesh models and data assimilation techniques in complex reservoirs

We present a methodology that allows conditioning the spatial distribution of geological and petrophysical properties of reservoir model realizations on available production data. The approach is fully consistent with modern concepts depicting natural reservoirs as composite media where the distribution of both lithological units (or facies) and associated attributes are modeled as stochastic processes of space. We represent the uncertain spatial distribution of the facies through a Markov mesh (MM) model, which allows describing complex and detailed facies geometries in a rigorous Bayesian framework. The latter is then embedded within a history matching workflow based on an iterative form of the ensemble Kalman filter (EnKF). We test the proposed methodology by way of a synthetic study characterized by the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm and its ability with respect to the standard EnKF to properly estimate model parameters and assess future reservoir production. We show the feasibility of integrating MM in a data assimilation scheme. Our methodology is conducive to a set of updated model realizations characterized by a realistic spatial distribution of facies and their log permeabilities. Model realizations updated through our proposed algorithm correctly capture the production dynamics.     

Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with EnKF

In a previous paper, we developed a theoretical basis for parameterization of reservoir model parameters based on truncated singular value decomposition (SVD) of the dimensionless sensitivity matrix. Two gradient-based algorithms based on truncated SVD were developed for history matching. In general, the best of these “SVD” algorithms requires on the order of 1/2 the number of equivalent reservoir simulation runs that are required by the limited memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm. In this work, we show that when combining SVD parameterization with the randomized maximum likelihood method, we can achieve significant additional computational savings by history matching all models simultaneously using a SVD parameterization based on a particular sensitivity matrix at each iteration. We present two new algorithms based on this idea, one which relies only on updating the SVD parameterization at each iteration and one which combines an inner iteration based on an adjoint gradient where during the inner iteration the truncated SVD parameterization does not vary. Results generated with our algorithms are compared with results obtained from the ensemble Kalman filter (EnKF). Finally, we show that by combining EnKF with the SVD-algorithm, we can improve the reliability of EnKF estimates.     

An iterative ensemble Kalman filter for reservoir engineering applications

The study has been focused on examining the usage and the applicability of ensemble Kalman filtering techniques to the history matching procedures. The ensemble Kalman filter (EnKF) is often applied nowadays to solving such a problem. Meanwhile, traditional EnKF requires assumption of the distribution’s normality. Besides, it is based on the linear update of the analysis equations. These facts may cause problems when filter is used in reservoir applications and result in sampling error. The situation becomes more problematic if the a priori information on the reservoir structure is poor and initial guess about the, e.g., permeability field is far from the actual one. The above circumstance explains a reason to perform some further research concerned with analyzing specific modification of the EnKF-based approach, namely, the iterative EnKF (IEnKF) scheme, which allows restarting the procedure with a new initial guess that is closer to the actual solution and, hence, requires less improvement by the algorithm while providing better estimation of the parameters. The paper presents some examples for which the IEnKF algorithm works better than traditional EnKF. The algorithms are compared while estimating the permeability field in relation to the two-phase, two-dimensional fluid flow model.