Geir Nævdal

Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter

The use of ensemble Kalman lter techniques for continuous updating of reservoir model is demonstrated. The ensemble Kalman lter technique is introduced, and thereafter applied on two 2-D reservoir models. One is a synthetic model with two producers and one injector. The other model is a simplied 2-D eld model, which is generated by using a horizontal layer from a North Sea eld model. By assimilating measured production data, the reservoir model is continuously updated. The updated models give im- proved forecasts. Both dynamic variables, as pressure and sat- urations, and static variables as the permeability are updated in the reservoir model. used to update static parameters in near-well reservoir models, by tuning the permeability eld. In this paper, the lter has been further developed to tune the permeability for simplied real eld reservoir simulation models. We present results from a synthetic model as well as a sim- plied real eld model. The measurements are well bottom- hole pressures, water cuts and gas/oil ratios. A synthetic model gives the possibility of comparing the solution obtained by the lter to the true solution, and the performance of the lter can be evaluated. It is shown how the reservoir model is updated as new measurements becomes available, and that good fore- casts are obtained. The convergence of the reservoir properties to the true solution as more measurements becomes available is investigated. Since the members of the ensemble are updated indepen- dently of each other, the method is very suitable for parallel processing. It is also conceptually straightforward to extend the methodology to update other reservoir properties than the per- meability. Based on the updated ensemble of models, production fore- casts and reservoir management studies may be performed on a single iaveragei model, which is always consistent with the latest measurements. Alternatively, the entire ensemble may be applied to estimate the uncertainties in the forecasts.

Adaptive Multiscale Permeability Estimation

With multiscale permeability estimation one does not select parameterization prior to the estimation. Instead, one performs a hierarchical search for the right parameterization while solving a sequence of estimation problems with an increasing parameterization dimension. In some previous works on the subject, the same refinement is applied all over the porous medium. This may lead to over-parameterization, and subsequently, to unrealistic permeability estimates and excessive computational work. With adaptive multiscale permeability estimation, the new parameterization at an arbitrary stage in the estimation sequence is such that new degrees of freedom are not necessarily introduced all over the porous medium. The aim is to introduce new degrees of freedom only where it is warranted by the data. In this paper, we introduce a novel adaptive multiscale estimation. The approach is used to estimate absolute permeability from two-phase pressure data in several numerical examples.

An Iterative Ensemble Kalman Filter

The ensemble Kalman filter is a Monte Carlo method for state estimation of nonlinear models, developed as an alternative or improvement of the extended Kalman filter. In this technical note we introduce an iterative extension to the ensemble Kalman filter. Iterations are introduced to improve the estimates in the cases where the relationship between the model and observations is not linear. The iterations converge, but to a solution where the data are overfitted. An essential stopping criteria is therefore introduced for the proposed method.

History Matching Using the Ensemble Kalman Filter on a North Sea Field Case

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Abstract Recently, the ensemble Kalman filter (EnKF) has been examined in several synthetic cases as an alternative to traditional history matching methods. Results from these studies indicate that the method can be useful for estimation of permeability and porosity fields. Contrary to other history matching methods, the EnKF provides an ensemble of model realizations containing information of the uncertainty in the estimates. Moreover, the data is processed sequentially, which makes it possible to always have an updated model conditioned on the most recent production data. The method therefore seems promising for real time reservoir management. This paper presents a successful study for a North Sea field case, where real production data have been assimilated using EnKF.

Analysis of the ensemble Kalman filter for estimation of permeability and porosity in reservoir models

It has lately been reported several successful applications where the ensemble Kalman filter has been used to estimate reservoir properties such as permeability and porosity. However, a thorough investigation of robustness and performance is still missing for this approach. In this paper we aim at filling this gap by studying the robustness of the methodology. One aspect which is investigated is how the filter depends on the initial ensemble. As the initial ensemble is created in a stochastic way, one can not be certain that the results obtained from one run represent the filter performance. Another aspect of interest is how prior information can be used to obtain best possible initial fields. The influence of geostatistical information on the estimated solutions is studied. In addition, the quality of the estimated fields is investigated by evaluating if the estimated static fields are reasonable when treated as the solution of the history matching problem. The estimation technique has been applied to the widely used PUNQ-S3 reservoir model, which is a small size synthetic 3-D reservoir engineering model. Both permeability and porosity are tuned, and measurements consist of well bottom-hole pressures, water cuts and gas-oil ratios. The initial fields are conditioned on the porosities in the gridblocks where the wells are located. By using a synthetic reservoir model it is possible to calculate the uncertainty of forecasts, and compare this with the true solution.

History Matching and Production Forecast Uncertainty by Means of the Ensemble Kalman Filter: A Real Field Application

During history match reservoir models are calibrated against production data to improve forecasts reliability. Often, the calibration ends up with a handful of matched models, sometime achieved without preserving the prior geological interpretation. This makes the outcome of many history matching projects unsuitable for a probabilistic approach to production forecast, then motivating the quest of methodologies casting history match in a stochastic framework. The Ensemble Kalman Filter (EnKF) has gained popularity as Monte-Carlo based methodology for history matching and real time updates of reservoir models. With EnKF an ensemble of models is updated whenever production data are available. The initial ensemble is generated according to the prior model, while the sequential updates lead to a sampling of the posterior probability function. This work is one of the first to successfully use EnKF to history match a real field reservoir model. It is, to our knowledge, the first paper showing how the EnKF can be used to evaluate the uncertainty in the production forecast for a given development plan for a real field model. The field at hand was an on-shore saturated oil reservoir. Porosity distribution was one of the main uncertainties in the model, while permeability was considered a porosity function. According to the geological knowledge, the prior uncertainty was modeled using Sequential Gaussian Simulation and ensembles of porosity realizations were generated. Initial sensitivities indicated that conditioning porosity to available well data gives superior results in the history matching phase. Next, to achieve a compromise between accuracy and computational efficiency, the impact of the size of the ensemble on history matching, porosity distribution and uncertainty assessment was investigated. In the different ensembles the reduction of porosity uncertainty due to production data was noticed. Moreover, EnKF narrowed the production forecast confidence intervals with respect to estimate based on prior distribution. Introduction Reservoir management of modern oil and gas fields requires periodic updates of the simulation models to integrate in the geological parameterization production data collected over time. In these processes the challenges nowadays are many. First, a coherent view of the geomodel requires updating the simulation decks in ways consistent with geological assumptions. Second, the management is requiring more and more often a probabilistic assessment of the different development scenarios. This means that cumulative distribution functions, reflecting the underlying uncertainty in the knowledge of the reservoir, for key production indicators, e.g. cumulative oil production at Stock Tank condition (STC), along the entire time-life of the field, are expected outcomes of a reservoir modeling project. Moreover, production data are nowadays collected with increasing frequencies, especially for wells equipped with permanent down-hole sensors. Decision making, based on most current information, requires frequent and rapid updates of the reservoir models. The Ensemble Kalman Filter (EnKF) is a Monte-Carlo based method developed by Evensen to calibrate oceanographic models by sequential data assimilation. Since the pioneering application on near-well modeling problems by Naevdal et al., EnKF has become in the reservoir simulation community a popular approach for history matching and uncertainty assessment. This popularity is motivated by key inherent features of the method. EnKF is a sequential data assimilation methodology, and then production data can be integrated in the simulation model as they are available. This makes EnKF well suited for realtime application, where data continuously collected have to be used to improve the reliability of predictive models. EnKF maintains a Gaussian ensemble of models aligned with the most current production data by linear updates of the model parameters. In that way the statistical properties of the Gaussian ensemble, that is to say mean, variance and twopoint correlations are preserved. Because EnKF does not need either history matching gradients or sensitivity coefficients, any reservoir simulator with restarting capabilities can be used in an EnKF workflow, without modifying simulator source code. This represents an obvious advantage with respect to methods like the Randomized Maximum Likelihood (RML) method, which requires a simulator with adjoint gradient capabilities. These reasons motivate the interest on EnKF in the Upstream Industry. Nonetheless, only a few real applications were published before this work. Skjervheim et. al. compared results on using EnKF to assimilate 4D seismic data and production data, and obtained results that slightly improved the base case used for comparison. Haugen et al., see Ref. 13, report that the EnKF was used to successfully history match the simulation model of a Northern sea field, with substantial improvement compared to the reference case. In this paper we applied EnKF to history match the Zagor simulation model, quantifying also the reduction of uncertainty due to the assimilation of the production data. Different ensembles were used to investigate the connection between the effectiveness of EnKF and the size of the statistical samples. Next, we used one of the ensembles updated with EnKF to assess the uncertainty in the production forecasts. To our knowledge, this is the first paper where EnKF was used on a real reservoir from history match to uncertainty analysis of production forecasts. The paper proceeds as follows. The next section is dedicated to the discussion of the EnKF methodology, including its mathematical background and some remarks on the current limitations. Then the Zagor reservoir model is described. That includes the geological parameterization used in this work and the presentation of the different ensembles utilized in the application. The results of the application are presented in two subsequent sections. The first is dedicated to history matching and the second dedicated to the assessment of the uncertainty in the production forecasts. Finally, conclusions based on our results are drawn and some perspectives for future works are given. The Ensemble Kalman Filter The EnKF is a statistical methodology suitable to solve inverse problem, especially in cases where observed data are available sequentially in time. Assuming that the evolution of a physical system can be approximated by a numerical model, typically by the discretisation of a partial differential equation, a state vector can be used to represent the model parameters and observations. Using multiple realizations of the state vector one is able to explicitly express the model uncertainty. The EnKF can describe the evolution of the system by updating the ensemble of state vectors whenever an observation is available. In reservoir simulation, EnKF can be applied to integrate production data by updating sequentially an ensemble of reservoir models during the simulation. Each reservoir model in the ensemble is kept up-to-date as production data are assimilated sequentially. In this context every reservoir state vector comprises three types of parameters: static parameters, dynamic parameters and production data. The static parameters are the parameters that in traditional history matching do not vary with time during a simulation, such as permeability (K) and porosity (φ). The dynamic parameters include the fundamental variables of the flow simulation. These are, for black oil models, the cell pressure (p), water saturation (Sw), gas saturation (Sg) and solution gas-oil ratio (RS). In addition to the variables for each cells one add observations of the production data in each well. Production data usually include simulated data corresponding to observations such as well production rates, bottom-hole pressure values, water cut (WCT) and gas oil ratio (GOR) values. Thus, using the notation by X. H. Wen and W. H. Chen, the ensemble of state variables is modelled by multiple realizations:

Iterative Ensemble Smoother as an Approximate Solution to a Regularized Minimum-Average-Cost Problem: Theory and Applications

The focus of this work is on an alternative implementation of the iterative ensemble smoother (iES). We show that iteration formulae similar to those used in \cite{chen2013-levenberg,emerick2012ensemble} can be derived by adopting a regularized Levenberg-Marquardt (RLM) algorithm \cite{jin2010regularized} to approximately solve a minimum-average-cost (MAC) problem. This not only leads to an alternative theoretical tool in understanding and analyzing the behaviour of the aforementioned iES, but also provides insights and guidelines for further developments of the smoothing algorithms. For illustration, we compare the performance of an implementation of the RLM-MAC algorithm to that of the approximate iES used in \cite{chen2013-levenberg} in three numerical examples: an initial condition estimation problem in a strongly nonlinear system, a facies estimation problem in a 2D reservoir and the history matching problem in the Brugge field case. In these three specific cases, the RLM-MAC algorithm exhibits comparable or even better performance, especially in the strongly nonlinear system.

Modelling two-phase flow for control design in oil well drilling

Today, marginal oil wells are being drilled and the operating margins for the bottom-hole well pressures during drilling are becoming narrower. This requires an improved control of the pressure balance between the reservoir pore pressure and the well bottom-hole pressure. In oil well drilling applications, the pressure is typically controlled manually by adjusting the choke valve. This paper proposes a simple feedback PI-control scheme with feed-forward compensation of the known disturbances. A low-dimensional dynamic state model for two-phase flow has been developed to be able to tune the control parameters. The proposed method is presented and evaluated using a detailed oil well drilling simulator. The results show that the proposed control design keeps the bottom-hole pressure within the operating margins

Multiscale estimation with spline wavelets, with application to two-phase porous-media flow

We consider the inverse problem of recovery of unknown coefficient functions in differential equations. The set of PDEs constituting the current forward model describes a special case of two-phase porous-media flow. The focus of the paper is on the influence of different length scales on parameter estimation efficiency. The investigation into these issues is facilitated by applying a multiscale spline wavelet parametrization of the unknown function. Earlier investigations with an ODE forward model found that use of the multiscale Haar parametrization had a positive effect on the estimation efficiency of a quasi-Newton algorithm. Recently, a way to systematically enhance these effects has been suggested. In this paper, we further this approach with the Levenberg-Marquardt algorithm. This results in three variants of the Levenberg-Marquardt algorithm, each incorporating a possibility to enhance multiscale effects. Through numerical experiments with the PDE forward model, we assess the estimation efficiency of the variants when varying the enhancement of multiscale effects.

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.