Laura Dovera

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:

Ensemble Kalman Filter for Gaussian Mixture Models

The Ensemble Kalman Filter (EnKF) is a statistical method to update dynamic models by sequential data assimilation. Recently EnKF has gained popularity in the reservoir simulation community as efficient history matching tool. The validity of EnKF update equations relies on the analytical solution of linear inverse problem with Gaussian prior. This assumption is critical in dealing with reservoir facies models. Variables associated to facies are better represented by multimodal distributions than normal priors which are used in the EnKF update scheme. In this paper we propose to model multimodal variables related to facies by Gaussian Mixture Models (GMM) and to modify EnKF for updating Gaussian Mixture (GM) distributions. First we derived the posterior distribution for a linear inverse problem assuming GM priors and the analytical solution we obtained shows that this posterior is again a GM. Using this result we then revisited the EnKF updating and we reformulated the update equations when the priors is assumed to be a GMM. We show two simple examples that give evidence of a good flexibility of GMM in managing multimodal distributions even though some computational issues linked to large scale applications are worth of a deeper investigation.

Sequential Simulations of Mixed Discrete-Continuous Properties: Sequential Gaussian Mixture Simulation

We present here a method for generating realizations of the posterior probability density function of a Gaussian Mixture linear inverse problem in the combined discrete-continuous case. This task is achieved by extending the sequential simulations method to the mixed discrete-continuous problem. The sequential approach allows us to generate a Gaussian Mixture random field that honors the covariance functions of the continuous property and the available observed data. The traditional inverse theory results, well known for the Gaussian case, are first summarized for Gaussian Mixture models: in particular the analytical expression for means, covariance matrices, and weights of the conditional probability density function are derived. However, the computation of the weights of the conditional distribution requires the evaluation of the probability density function values of a multivariate Gaussian distribution, at each conditioning point. As an alternative solution of the Bayesian inverse Gaussian Mixture problem, we then introduce the sequential approach to inverse problems and extend it to the Gaussian Mixture case. The Sequential Gaussian Mixture Simulation (SGMixSim) approach is presented as a particular case of the linear inverse Gaussian Mixture problem, where the linear operator is the identity. Similar to the Gaussian case, in Sequential Gaussian Mixture Simulation the means and the covariance matrices of the conditional distribution at a given point correspond to the kriging estimate, component by component, of the mixture. Furthermore, Sequential Gaussian Mixture Simulation can be conditioned by secondary information to account for non-stationarity. Examples of applications with synthetic and real data, are presented in the reservoir modeling domain where realizations of facies distribution and reservoir properties, such as porosity or net-to-gross, are obtained using Sequential Gaussian Mixture Simulation approach. In these examples, reservoir properties are assumed to be distributed as a Gaussian Mixture model. In particular, reservoir properties are Gaussian within each facies, and the weights of the mixture are identified with the point-wise probability of the facies.

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.

Improved initial ensemble generation coupled with ensemble square root filters and inflation to estimate uncertainty

The performance of the Ensemble Kalman Filter method (EnKF) depends on the sample size compared to the dimension of the parameters space. In real applications insufficient sampling may result in spurious correlations which reduce the accuracy of the filter with a strong underestimation of the uncertainty. Covariance localization and inflation are common solutions to these problems. The Ensemble Square Root Filters (ESRF) is also better to estimate uncertainty with respect to the EnKF. In this work we propose a method that limits the consequences of sampling errors by means of a convenient generation of the initial ensemble. This regeneration is based on a Stationary Orthogonal-Base Representation (SOBR) obtained via a singular value decomposition of a stationary covariance matrix estimated from the ensemble. The technique is tested on a 2D single phase reservoir and compared with the other common techniques. The evaluation is based on a reference solution obtained with a very large ensemble (one million members) which remove the spurious correlations. The example gives evidence that the SOBR technique is a valid alternative to reduce the effect of sampling error. In addition, when the SOBR method is applied in combination with the ESRF and inflation, it gives the best performance in terms of uncertainty estimation and oil production forecast.

Integration of Markov Mesh Models and Ensemble Data Assimilation in Reservoirs with Complex Geology

We present a methodology conducive to updating both facies and petrophysical properties of a reservoir models set characterized by a complex architecture within the context of a history matching procedure based on the Ensemble Kalman Filter (EnKF). Spatial distribution of facies is handled by means of a Markov Mesh (MM) model. The latter is adopted because of its ability to reproduce detailed facies geometries and spatial patterns and can be integrated in a consistent probabilistic Bayesian framework. The MM model is then integrated into a history matching procedure which is based on the EnKF scheme. We test the proposed methodology by means of a synthetic reservoir in the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm with respect to the standard EnKF both in terms of history matching quality and forecast prediction capabilities. The results show that the integration of MM in the data assimilation scheme allows obtaining realistic geological shapes for spatial facies distribution and an improved estimation of petrophysical properties. In addition, the updated ensemble correctly captures the production range in the long term.

Improved Initial Ensemble Generation Coupled with Ensemble Square Root Filters and Boosting to Estimate Uncertainty

The accuracy of ensemble Kalman filter (EnKF) methods depends on the sample size compared to the dimension of the parameters space. In real applications often sampling error may result in spurious correlations which produce a bias in the mean and a strong underestimation of the uncertainty. The Ensemble Square Root Filters (ESRF) represents an advantage in uncertainty estimation respect to the traditional EnKF. Covariance inflation and localization are a common solution to these problems. In this work we propose a method that reduces the bias of ensemble techniques by means of a convenient generation of the initial ensemble. This regeneration is based on a Stationary Orthogonal-Base Representation (SOBR), obtained via a singular value decomposition of a stationary covariance matrix estimated from the ensemble. This technique is tested on a 2D slightly compressible single phase model and compared with ESRF. The comparison is based on a reference solution obtained with a very large ensemble (one million). The example gives evidence that the SOBR reduces the effect of sampling error in the mean but covariance inflation is essential to avoid the ensemble collapse.

STATE ESTIMATION OF A LARGE-SCALE SYSTEM IN THE PETROLEUM INDUSTRY: THE ENSEMBLE KALMAN FILTER FOR UPDATING RESERVOIR MODELS

Abstract The ensemble Kalman filter is presented and it is shown how it is applied for updating models for fluid flow in oil reservoirs. A set of updated models are produced, which fit better to the available measurements and will be useful for further decision making.