Application and Improvement of Ensemble Kalman Filter Method in Production Data Analysis

A localized ensemble Kalman filter (EnKF) method is developed to assimilate transient flow data to calibrate a heterogeneous conductivity field. To update conductivity value at a point in a study domain, instead of assimilating all the measurements in the study domain, only limited measurement data in an area around the point are used for the conductivity updating in the localized EnKF method. The localized EnKF is proposed to solve the problems of the filter divergence usually existing in a data assimilation method without localization. The developed method is applied, in a synthetical two dimensional case, to calibrate a heterogeneous conductivity field by assimilating transient hydraulic head data. The simulations by the data assimilation with and without localized EnKF are compared. The study results indicate that the hydraulic conductivity field can be updated efficiently by the localized EnKF, while it cannot be by the EnKF. The covariance inflation and localization are found to solve the problem of the filter divergence efficiently. In comparison with the EnKF method without localization, the localized EnKF method needs smaller ensemble size to achieve stabilized results. The simulation results by the localized EnKF method are much more sensitive to conductivity correlation length than to the localization radius. The developed localized EnKF method provides an approach to improve EnKF method in conductivity calibration.

The adjoint and tangent linear models in the traditional four‐dimensional variational data assimilation (4DVAR) are difficult to obtain if the forecast model is highly nonlinear or the model physics contains parameterized discontinuities. A new method (referred to as POD‐E4DVAR) is proposed in this paper by merging the Monte Carlo method and the proper orthogonal decomposition (POD) technique into 4DVAR to transform an implicit optimization problem into an explicit one. The POD method is used to efficiently approximate a forecast ensemble produced by the Monte Carlo method in a 4‐dimensional (4‐D) space using a set of base vectors that span the ensemble and capture its spatial structure and temporal evolution. After the analysis variables are represented by a truncated expansion of the base vectors in the 4‐D space, the control (state) variables in the cost function appear explicit so that the adjoint model, which is used to derive the gradient of the cost function with respect to the control variables in the traditional 4DVAR, is no longer needed. The application of this new technique significantly simplifies the data assimilation process and retains the two main advantages of the traditional 4DVAR method. Assimilation experiments show that this ensemble‐based explicit 4DVAR method performs much better than the traditional 4DVAR and ensemble Kalman filter (EnKF) method. It is also superior to another explicit 4DVAR method, especially when the forecast model is imperfect and the forecast error comes from both the noise of the initial field and the uncertainty in the forecast model. Computational costs for the new POD‐E4DVAR are about twice as the traditional 4DVAR method but 5% less than the other explicit 4DVAR and much lower than the EnKF method. Another assimilation experiment conducted within the Lorenz model indicates potential wider applications of this new POD‐E4DVAR method.

Summary The concept of "closed-loop" reservoir management is currently receiving considerable attention in the petroleum industry. A "realtime" or "continuous" reservoir model updating technique is a critical component for the feasible application of any closed-loop, model-based reservoir management process. This technique should be able to rapidly and continuously update reservoir models assimilating the up-to-date observations of production data so that the performance predictions and the associated uncertainty are up-to-date for optimization of future development/operations. The ensemble Kalman filter (EnKF) method has been shown to be quite efficient for this purpose in large-scale nonlinear systems. Previous studies show that a relatively large ensemble size is required for EnKF to reliably assess the uncertainty, and a confirming step is recommended to ensure the consistency of the updated static and dynamic variables with the flow equations. In this paper, we further explore the capability of EnKF, focusing on some practical issues including the correction of the linear and Gaussian assumptions during filter updating with iteration, the reduction of ensemble size with a resampling scheme, and the impact of data assimilation time interval. Results from the example in this paper demonstrate that the proposed iterative EnKF performs better with more accurate predictions and less uncertainty than the traditional noniterative EnKF. The use of iteration reduces the impact of nonlinearity and non-Gaussianity. Results also show that iteration may only be required when predictions are considerably deviated from the observations. The proposed resampling scheme can significantly reduce the ensemble size necessary for reliable assessment of uncertainty with improved accuracy. Finally, we show that the noniterative EnKF is sensitive to the size of time interval between the assimilation steps. Using the proposed iterative EnKF, results are more stable, more accurate reservoir models and predictions can be obtained even when a large time interval is used. This also indicates that iteration within the EnKF updating serves as a process that corrects the stronger nonlinear and non-Gaussian behaviors when larger time interval is used.

A new sequential data assimilation method named “Monte Carlo H∞ filter” is introduced based on H∞ filter technique and Monte Carlo method in this paper. This method applies to nonlinear systems in condition of lacking the statistical properties of observational errors. In order to compare the assimilation capability of Monte Carlo H∞ filter with that of the ensemble Kalman filter (EnKF) in solving practical problems caused by temporal correlation or spatial correlation of observational errors, two numerical experiments are performed by using Lorenz (1963) system and shallow-water equations respectively. The result is that the assimilation capability of the new method is better than that of EnKF method. It is also shown that Monte Carlo H∞ filter assimilation method is effective and suitable to nonlinear systems in that it does not depend on the statistical properties of observational errors and has better robustness than EnKF method when the statistical properties of observational errors are varying. In addition, for the new method, the smallest level factor founded by search method is flow-dependent.

Saturation-based localization within EnKF approach for history matching in SAGD processes

An ensemble Kalman filter (EnKF) method is proposed to track dynamic states of generators. The algorithm of the EnKF and its application to generator state tracking are presented in detail. The accuracy and sensitivity of the method are analyzed with respect to initial state errors, measurement noise, unknown fault locations, time steps and parameter errors. It is demonstrated through simulation studies that even with some errors in the parameters, the developed EnKF method can still effectively track generator dynamic states.

This paper presents the application of the ensemble Kalman filter (EnKF) method to the integration of well test data into heterogeneous reservoir models generated from geological and geophysical data. EnKF does not require computing the gradient of an objective function, and hence can be applied easily with any reservoir simulator, and more importantly, is far more efficient than a gradient-based history matching procedure when the forward model is represented by a reservoir simulator. In the procedure for integrating pressure transient data considered here, the static geological/geophysical data are assumed to be encapsulated in a multivariate probability density function characterized by a prior mean and covariance for the joint distribution of the porosity and permeability fields. As the prior mean of the property fields obtained from the core and log data can sometimes be erroneous, a partially doubly stochastic model is applied to account for the uncertainty of the prior mean. In the doubly stochastic model, a correction to the prior mean is adjusted together with the heterogeneous field during history matching. The method is tested with synthetic heterogeneous single-layer and two-layer reservoirs. Excellent data matches are obtained with EnKF in a small fraction of the time that would be required for a gradient-based history-matching process and the observed data fall within the uncertainty bounds of the ensemble data predictions. In the layered reservoir case, the uncertainty in rock property fields is further reduced by integration of production log data (layer flow rates). We show that problems of statistical inconsistency and poor data matches that are sometimes encountered when matching production data with EnKF do not occur when matching pressure data with the EnKF implementation used here. This undoubtedly occurs because the dynamical system (reservoir simulator equations) is much more nearly linear for the single-phase flow problems considered here than for multiphase flow cases.

Ensemble Kalman filter for automatic history matching of geologic facies

Data assimilation is the process of estimating the state of dynamic systems (linear or nonlinear, Gaussian or non-Gaussian) as accurately as possible from noisy observational data. Although the Three Dimensional Variational (3D-VAR) methods, Four Dimensional Variational (4D-VAR) methods and Ensemble Kalman filter (EnKF) methods are widely used and effective for linear and Gaussian dynamics, new methods of data assimilation are required for the general situation, that is, nonlinear non-Gaussian dynamics. General Bayesian recursive estimation theory is reviewed in this thesis. The Bayesian estimation approach provides a rather general and powerful framework for handling nonlinear, non-Gaussian, as well as linear, Gaussian estimation problems. Despite a general solution to the nonlinear estimation problem, there is no closed-form solution in the general case. Therefore, approximate techniques have to be employed. In this thesis, the sequential Monte Carlo (SMC) methods, commonly referred to as the particle filter, is presented to tackle non-linear, non-Gaussian estimation problems. In this thesis, we use the SMC methods only for the nonlinear state estimation problem, however, it can also be used for the nonlinear parameter estimation problem. In order to demonstrate the new methods in the general nonlinear non-Gaussian case, we compare Sequential Monte Carlo (SMC) methods with the Ensemble Kalman Filter (EnKF) by performing data assimilation in nonlinear and non-Gaussian dynamic systems. The models used in this study are referred to as state-space models. The Lorenz 1963 and 1966 models serve as test beds for examining the properties of these assimilation methods when used in highly nonlinear dynamics. The application of Sequential Monte Carlo methods to different fixed parameters in dynamic models is considered. Four different scenarios in the Lorenz 1063 [sic] model and three different scenarios in the Lorenz 1996 model are designed in this study for both the SMC methods and EnKF method with different filter siz

The Horseshoe Canyon (HSC) CBM play of the Western Canadian Sedimentary Basin is unique to low-rank coal reservoirs because of lack of water production; the production characteristics are qualitatively similar to conventional low-pressure dry gas reservoirs. However, the complex geologic history of the coals and non-coal interbeds has imparted strong vertical and lateral heterogeneities that make the play difficult to characterize using conventional methods. For example, in a typical HSC well, there are often > 10 seams to complete, which may exhibit strong contrasts in initial pressure, gas content, thickness and permeability. Further, the lateral continuities of the individual seams vary, and are often not correlatable from well-to-well. Recently, advances in production data analysis (PDA) methodologies have been made for CBM wells; techniques developed for tight gas, and conventional oil and gas reservoirs have been adapted by incorporating some CBM reservoir properties. For example, the popular flowing material balance (FMB) technique, as well as production type-curve and pressure transient analysis (PTA) have been modified to include relatively simple CBM reservoir behavior (ex. equilibrium desorption). These methods, however, are primarily restricted to the analysis of single-layer reservoirs; significant errors in estimates of original-gas-in-place (OGIP) and other reservoir properties may occur if strong contrasts exist from layer-to-layer. In this work, multi-layer analysis tools are discussed, including analytical simulators that are used to history-match layer-allocated rates and pressures, and layer-specific FMB, which is used as a PDA method for individual layers. A study was undertaken to establish the applicability of advanced PDA methods to the quantitative assessment of HSC reserves. Single-layer and multi-layer analysis tools were first tested against artificial data created with a numerical simulator. Next, single-layer-equivalent analysis was performed on > 40 real wells using type-curve, FMB, and analytical simulation. Many of the wells exhibit an early "cleanup" period with inclining or flat production, precluding analysis of transient flow data. Finally, a more rigorous multi-layer analysis was performed on a subset of wells where spinner surveys, and individual-seam pressure buildup data, collected at several points in time during the producing life of the well, were available. Analysis of this subset of wells included PTA of the individual seams, individual seam material balance, and multi-layer (up to 6) analytical simulation history-matching of total commingled flow rates, individual coal zone rates estimated from spinner surveys, and shut-in pressures. The resulting reservoir property estimates (ex. total OGIP) were then compared to the single-layer-equivalent analysis; the single-layer-equivalent analysis appears to yield conservative estimates of OGIP. Future work will include continued comparisons of multi-layer vs. single-layer PDA, development of new PDA methods for analyzing multi-layer reservoirs, investigation of additional constraints on permeability and other reservoir properties that can be used in multi-layer history-matching process, and time-lapse PTA work to quantify changes in layer permeability and skin during depletion.

The Ensemble Kalman Filter (EnKF) has gained popularity over recent years as a Monte-Carlo based technique for assisted history matching and real time updating of reservoir models. The EnKF procedure utilizes an ensemble of model states (e.g. realizations of reservoir properties such as porosity and permeability) to approximate the covariance matrices used in the updating process. EnKF works efficiently with Gaussian variables and linear dynamics, but it often fails to preserve the reference probability distribution of the model parameters and to achieve an acceptable production data match where the system dynamics are strongly nonlinear, especially of the type related to multiphase flow, or if non-Gaussian prior models are used. In order to alleviate these drawbacks, we investigated various weighted averaging techniques for computing the ensemble mean by introducing a weighting factor to each ensemble; two new formulations were implemented. The first weighting factor was calculated based on the mismatch in entropy of the model parameters, a normalized measure of the spread of a given probability distribution. The second weighting factor was computed using the forecast mismatch. In addition, both weights could be applied at a single updating step for reducing the forecast mismatch and maintaining the prior distribution simultaneously. The performance of traditional EnKF and these weighted EnKF methods were evaluated by performing various simulation studies with different reservoir heterogeneity. The qualities of the final matching results were assessed by computing the experimental histogram and variograms of the final ensemble, as well as the Root Mean Square Error (RMSE) of the predicted data mismatch. The results reveal that reasonable improvement in the efficiency of the EnKF is achieved by suggested weighted techniques. The RMSE of the predicted data is improved, and the quantity of spurious model parameters is reduced at each updating step. Taking advantage of the entropy based weighting factor assists the filter to preserve the reference distribution. The improvement indicates that the Entropy weighted EnKF (EWEnKF) has a significant potential to resolve the shortfall of traditional EnKF in reservoir characterization and history matching of challenging reservoirs with non-Gaussian distributions.

First and foremost, production analysis techniques require accurate rate and bottomhole pressure histories. In most cases the pressure history of the well is not measured directly at the bottomhole condition, but is calculated from surface measurements by the use of single or multiphase flow correlations. In some cases significant error is introduced through the use of these correlations. This paper evaluates the magnitude of such errors for oil and gas producers with regard to the estimation of flow capacity, completion efficiency, and effective drainage area. Synthetic cases are used as control sets in order to evaluate the sensitivity of the results to the various multiphase flow correlations and flowing conditions. In addition to synthetic (simulated) performance behavior, field cases are presented and the variance in estimated reservoir and completion properties is evaluated. The technical contributions of this paper are: Systematic evaluation of the effect of errors in flow rates and bottomhole flowing pressures on production data analysis — using both synthetic and field derived well performance data. Qualitative guidelines as the effect of errors in rate and pressure on estimated reservoir properties. Establishment of the most relevant pressure drop correlations for use in practice.

A two-stage ensemble Kalman filter based on multiscale model reduction for inverse problems in time fractional diffusion-wave equations

One aspect of implementing a limited-area ensemble Kalman filter (EnKF) involves the specification of a suitable ensemble of lateral boundary conditions. Two classes of methods to populate a boundary condition ensemble are proposed. In the first class, the ensemble of boundary conditions is provided by an EnKF on a larger domain and is approximately a random draw from the probability distribution function for the forecast (or analysis) on the limited-area domain boundary. The second class perturbs around a deterministic estimate of the state using assumed spatial and temporal covariance relationships. Methods in the second class are relatively flexible and easy to implement. Experiments that test the utility of these methods are performed for both an idealized low-dimensional model and limited-area simulations using the Weather Research and Forecasting (WRF) model; all experiments employ simulated observations under the perfect model assumption. The performance of the ensemble boundary condition methods is assessed by comparing the results of each experiment against a control “global” EnKF that extends beyond the limited-area domain. For all methods tested, results show that errors for the limited-area EnKF are larger near the lateral boundaries than those from a control EnKF, but decay inside the limited-area domain so that errors there are comparable to the control case. The relatively larger errors near the boundaries in the limited-area EnKF originate from not assimilating observations outside the limited-area domain and, in the second class of methods, from deficiencies in boundary spatial and temporal covariances. Overall, these experiments suggest that for observation densities typical in numerical weather prediction models, ensemble boundary conditions can be specified in the absence of a global ensemble without significant penalty in the domain interior by perturbing around an ensemble mean.

Improvement of the seismic reservoir characterisation is performed by introducing an uncertainty connected to parameters in the rock physics model. Traditionally the Ensemble Kalman filter (EnKF) method has been used to estimate permeability and porosity.<br><br>In this paper we show that when seismic difference data are available also the lithology can be estimated, which is coupled to the effective bulk modulus via the rock physics model. Incorporation of inverted seismic difference data in the EnKF introduces a large amount of data in the assimilation step. Thus, to improve the results a methodology based on a combination of a global and a local analysis scheme is proposed. The global and the local analysis are used to assimilate respectively the production data and the inverted seismic difference data, where the local scheme assume that only seismic data within a certain distance from a state variable will impact the analysis in this state variable. The technique is applied to synthetic 2D and 3D reservoir models, where effects of using local versus global analysis schemes on different inverted seismic difference data, such as acoustic impedance and Poisson's ratio, are investigated. Other evaluated factors are the effects of using an incorrect seismic data error model in the analysis schemes.