Jonggeun Choe

Improvement of ensemble smoother with clustered covariance for channelized reservoirs

Ensemble Kalman filter (EnKF) has been researched for reservoir characterization in petroleum engineering. However, the repeated assimilation causes lots of simulation cost. Ensemble smoother (ES) assimilates all available data once. It has advantages over EnKF: efficiency and simplicity. The two ensemble methods are based on the same assumptions: Gaussian distribution and trust in the mean of all ensembles. Many researchers have pointed out that EnKF gives overshooting and filter divergence problems when the two key assumptions are not satisfied. This paper presents characterization of channel fields using ES with the concept of clustered covariance, especially improper ensemble design. The standard EnKF, ES, and the proposed method are applied to a 2D synthetic channel field with 200 ensembles. From distance-based clustering method, we separate initial ensembles into 10 groups based its on similarity. The proposed method uses 10 Kalman gains, since each cluster has own Kalman gain. They can represent ensembles properly by using similar ensembles instead of 200 different ensembles. For the channel fields, the standard EnKF and ES show overshooting and filter divergence problems. Updated permeability fields have extreme values and lose the continuity of channel stream. However, the proposed method manages those two problems and provides reasonable results. We can get future prediction with reliable uncertainty. The proposed method only requires about 5% of simulation time compared to EnKF, since it is based on ES. It can be applied to the characterization of channel fields, even though we have improper ensembles due to limited information.

Reservoir characterization using a streamline-assisted ensemble Kalman filter with covariance localization

Ensemble Kalman filter (EnKF) is a recursive data process algorithm that uses continuous updating. It has been proven that EnKF is an efficient method for data assimilation, uncertainty assessment, and large scale problems in many engineering fields. However, there are two common limitations-filter divergence and overshooting/undershooting. These are due to reduction of cross-covariance between model parameters and measurements. We propose a streamline-assisted ensemble Kalman filter (SL EnKF) that uses covariance localization according to the types of well and measurement data. This method enables selective updates of permeability, therefore, providing more reliable permeability field estimations than the standard EnKF without overshooting/undershooting or filter divergence. In addition, it gives efficient uncertainty evaluations by considering the performances of each ensemble member.

Characterization of channelized reservoir using ensemble Kalman Filter with clustered covariance

Ensemble Kalman filter (EnKF) has the limitation of applications for multi-point geostatistics because it assumes Gaussian random field. It also uses all ensembles to get covariance matrix, even though they have different permeability field each other, resulting in filter divergence. The proposed method suggests the concept of clustered covariance by grouping initial ensembles using a distance-based method. Hausdorff distance is used for calculating similarity between permeability fields and they are separated by k-means clustering. When EnKF is applied to a 2D channel field, it shows overshooting problem and mismatches the true production data. The proposed method gives better history matching and future performance prediction without overshooting problems. Furthermore, it shows stable results for sensitivity analyses over the number of total ensembles. The more accurate covariance is calculated by clustering, the better results are obtained.

Uncertainty Quantification of Channelized Reservoir Using Ensemble Smoother with Selective Measurement Data

Ensemble smoother (ES) assimilates all available dynamic data without iterations as global update. Therefore, ES is much faster than ensemble Kalman filter (EnKF), which uses recursive updates. Iterative concepts are introduced for ES to increase accuracy of history matching. However, they lose advantages of simulation time and cost over EnKF. We propose ES with selective use of observation data in assimilation to improve history matching results and to keep simulation time short. Three methods, EnKF with all data, ES with all data, and the proposed, are applied to 2D synthetic channelized reservoirs with a nine spot waterflooding. Ensemble-based methods interpret the reason of reduced oil production rate after water breakthrough as low permeability. In this research, we suggest selective measurement data to manage misinterpretable data for the ensemble-based methods. As a logical choice, oil production rate before water breakthrough and water cut after water breakthrough are used for assimilation. EnKF with all data cannot predict true performances of oil and water productions on each well. ES with all data shows severe overshooting and filter divergence problems, which are two typical problems in the ensemble-based methods. However, the proposed method overcomes the two problems and shows good history matching results. It provides reliable uncertainty quantification of reservoir performances for both each well production and field total productions. Simulation cost of the proposed method is about 2.2% of that of EnKF, which uses 45 times update. It has clear advantage over EnKF or iterative ES methods.

Modeling of solute transport in a single fracture using streamline simulation and experimental validation

Abstract Streamline simulations have been extensively used in petroleum engineering due to its computational speed and the freedom from numerical dispersion. This study applies streamline simulation to the modeling of solute transport in a single fracture and verifies the streamline method with experimental data. In order to model dispersive transport, a new term, the advection–dispersion ratio is employed, which is defined as the relative extent of advection to dispersion along streamlines. It is observed that the tracer breakthrough curves from the simulation match well with those from the experiments. In addition, the tracer displacement profiles from the simulation also show resemblance to those from the experiments. Simulations with various link transmissivity types result in no serious disparities. The distributions of time of flight and tracer breakthrough curves from the simulations using different link transmissivity types are much alike. Transport simulation is performed by allocating different advection–dispersion ratios along streamlines. Afterwards, the results are compared with the simulation result using single representative advection–dispersion ratio over the flow domain. Although streamlines actually have different advection–dispersion ratios, its effect is found to be not severe. Therefore, a representative advection–dispersion ratio can be used for modeling transport through the whole streamlines in a single fracture.

Reservoir Characterization Using an EnKF and a Non-parametric Approach for Highly Non-Gaussian Permeability Fields

Abstract An inverse scheme is developed for reservoir characterization using ensemble Kalman filter (EnKF) and non-parametric approach. EnKF has been studied by many researchers due to its novelties on recursive data processing, easy access to parallel processing, and quantifying uncertainties on its results. However, previous studies have shown poor characterization results with non-Gaussian permeability distributions. In this study, non-parametric approach is used to characterize permeability distribution with strong non-Gaussian characteristics. Normal score transformation is utilized to satisfy the Gaussian assumption of EnKF in the assimilation step. From the analyses of initial ensembles effects with non-Gaussian distributions, initial ensembles with higher similarity to the reference distribution would give more successful characterization results than those of less similarity. Additional improvement in reservoir characterization results is obtained by using normal score transformation.

Reservoir Characterization from Insufficient Static Data Using Gradual Deformation Method with Ensemble Kalman Filter

Abstract Reservoir characterization is critical in order to estimate reserves and optimize oil and gas production. Ensemble Kalman filter characterizes the spatial distribution of reservoir parameters using covariances between static and dynamic data. Ensemble Kalman filter can rapidly provide results reflecting its overall tendency of parameter distribution, but may not characterize them in detail because ensemble Kalman filter does not minimize an objective function. Gradual deformation method is one of the well known non-gradient-based methods, which guarantees a global minimum theoretically. However, if permeability samples available do not represent the whole distribution, it can be hard for gradual deformation method to reduce the objective function sufficiently. In this study, results of ensemble Kalman filter are utilized as input permeability fields in gradual deformation method so that the demerit of each method is compensated by the merit. Although samples from a reservoir have no information on extreme permeability values, the developed gradual deformation method model with ensemble Kalman filter reliably characterizes hidden extreme values.

An inverse system for incorporation of conditioning to pressure and streamline-based calibration.

A streamline-based history matching technique is employed to perform fast and efficient permeability identification and to integrate tracer data into an inverse model. To incorporate tracer data into the inverse model, a given tracer breakthrough curve is interpreted as cumulative breakthrough along independent streamlines. Permeabilities are modified along each streamline to match the tracer breakthrough curve. In this way, there is no explicit computation of sensitivity coefficients, nor any matrix inversion. However, this approach is incomplete by itself. Since the modifications occur along the streamlines, the identified permeability distribution is often incompatible with the actual permeability distribution. Thus, streamlines should be positioned correctly before the streamline-based method is applied. To accomplish this, geostatistical methods such as kriging and sequential Gaussian simulation (SGS) are implemented to provide an appropriate disposition of streamlines at the beginning of the inverse process. Then, permeabilities are iteratively calibrated in a conventional grid system to satisfy pressure and permeability observation data, and simultaneously modified along streamlines to match tracer data. The two independent optimization processes assist mutually and lead to stable convergence to a minimum. By applying the proposed inverse system to synthetic reference fields, it is observed that identified fields satisfactorily reproduce the permeability distribution of the reference fields. In addition, the pressure distributions of the identified and the reference fields are fairly alike, and the identified tracer breakthrough curves are well fitted to those of the reference fields. With regard to spatial patterns of transport behaviors, the streamlines of the identified fields show similar trajectories to those of the reference fields, and the time of flight distributions of the inversed fields are also analogous to those of the reference fields. The proposed inverse system is capable of estimating the future performance of a two-dimensional aquifer from a constrained number of permeability and pressure observation data accompanied by tracer data.

Stochastic optimization for global minimization and geostatistical calibration

Abstract This study proposes a stochastic optimization technique that uses a gradient-based method as the primary optimization method, as well as a geostatistical conditional simulation to perturb and calibrate parameters at every local minimum. If the optimization process is trapped at a local minimum due to the limitations of the gradient-based method, it generates equi-probable parameter fields using a geostatistical conditional simulation. Among the generated fields, the optimization process selects one that enables the objective function to be reduced below the value of that at the local minimum, and then reactivates the gradient-based optimization. In generating equi-probable parameter fields, a constrained number of points (noted as releasing points) are randomly selected, and spatially correlated values are generated at the releasing points, conditioned to optimum parameters at the local minimum. By applying the stochastic optimization to synthetic permeability fields, it is observed that an inversed permeability field reproduces not only global distribution but also local spatial variability of the reference fields. In addition, the pressure distributions of the inversed and the reference field were much alike. To investigate dynamic properties of the inversed field and the reference field, streamline simulation was performed on both fields. Streamlines of the inversed field showed similar trajectories to those of the reference field, and time of flight (TOF) distribution of the inversed field was analogous to that of the reference field. The stochastic optimization technique proposed in this paper enables an inverse process to converge to a global minimum while preserving geostatistical properties such as mean, standard deviation, and variogram of an original field. Therefore, the stochastic optimization will be efficient in predicting future performance of a field from constrained number of permeability and pressure observation data.

Covariance Matrix Localization Using Drainage Area in an Ensemble Kalman Filter

The Ensemble Kalman filter has been widely researched because of its availability of real-time updating of reservoir models and uncertainty quantification. There have been many studies to solve the two typical problems: overshooting and filter divergence, and to increase its accuracy. One of the methods is covariance localization, which excludes data having relatively low correlation between reservoir parameters and observations. This article proposes covariance matrix localization using a drainage area, which can be easily obtained by checking the direction of oil flow. In applications to synthetic reservoirs, the proposed method gives a better prediction of the permeability distribution by history matching and, therefore, future performances. It also provides reliable results in cases of small ensemble sizes.