Summary The dynamical equations for multiphase flow in porous media are highly nonlinear and the number of variables required to characterize the medium is usually large, often two or more variables per simulator gridblock. Neither the extended Kalman filter nor the ensemble Kalman filter is suitable for assimilating data or for characterizing uncertainty for this type of problem. Although the ensemble Kalman filter handles the nonlinear dynamics correctly during the forecast step, it sometimes fails badly in the analysis (or updating) of saturations. This paper focuses on the use of an iterative ensemble Kalman filter for data assimilation in nonlinear problems, especially of the type related to multiphase ow in porous media. Two issues are key:iteration to enforce constraints andensuring that the resulting ensemble is representative of the conditional pdf (i.e., that the uncertainty quantification is correct). The new algorithm is compared to the ensemble Kalman filter on several highly nonlinear example problems, and shown to be superior in the prediction of uncertainty. Introduction For linear problems, the Kalman filter is optimal for assimilating measurements to continuously update the estimate of state variables. Kalman filters have occasionally been applied to the problem of estimating values of petroleum reservoir variables (Eisenmann et al. 1994; Corser et al. 2000), but they are most appropriate when the problems are characterized by a small number of variables and when the variables to be estimated are linearly related to the observations. Most data assimilation problems in petroleum reservoir engineering are highly nonlinear and are characterized by many variables, often two or more variables per simulator gridblock. The problem of weather forecasting is in many respects similar to the problem of predicting future petroleum reservoir performance. The economic impact of inaccurate predictions is substantial in both cases, as is the difficulty of assimilating very large data sets and updating very large numerical models. One method that has been recently developed for assimilating data in weather forecasting is ensemble Kalman filtering (Evensen 1994; Houtekamer and Mitchell 1998; Anderson and Anderson 1999; Hamill et al. 2000; Houtekamer and Mitchell 2001; Evensen 2003). It has been demonstrated to be useful for weather prediction over the North Atlantic. The method is now beginning to be applied for data assimilation in groundwater hydrology (Reichle et al. 2002; Chen and Zhang 2006) and in petroleum engineering (Nævdal et al. 2002, 2005; Gu and Oliver 2005; Liu and Oliver 2005a; Wen and Chen 2006, 2007; Zafari and Reynolds 2007; Gao et al. 2006; Lorentzen et al. 2005; Skjervheim et al. 2007; Dong et al. 2006), but the applications to state variables whose density functions are bimodal has proved problematic (Gu and Oliver 2006). For applications to nonlinear assimilation problems, it is useful to think of the ensemble Kalman filter as a least squares method that obtains an averaged gradient for minimization not from a variational approach but from an empirical correlation between model variables (Anderson 2003; Zafari et al. 2006). In addition to providing a mean estimate of the variables, a Monte Carlo estimate of uncertainty can be obtained directly from the variability in the ensemble.

This paper reports the use of ensemble Kalman filter (EnKF) for automatic history matching. EnKF is a Monte Carlo method, in which an ensemble of reservoir state variables are generated and kept up-to-date as data are assimilated sequentially. The uncertainty of reservoir state variables is estimated from the ensemble at any time step. Two synthetic problems are selected to investigate two primary concerns with the application of the EnKF. The first concern is whether it is possible to use a Kalman filter to make corrections to state variables in a problem for which the covariance matrix almost certainly provides a poor representation of the distribution of variables. It is tested with a one-dimensional, two-phase waterflood problem. The water saturation takes large values behind the flood front, and small values ahead of the front. The saturation distribution is bimodal and is not well modeled by the mean and variance. The second concern is the representation of the covariance via a relatively small ensemble of state vectors may be inadequate. It is tested by a two-dimensional, two-phase problem. The number of ensemble members is kept the same as for the one-dimensional problem. Hence the number of ensemble members used to create the covariance matrix is far less than the number of state variables. We conclude that EnKF can provide satisfactory history matching results while requiring less computation work than traditional history matching methods.

This paper reports the use of the ensemble Kalman filter (EnKF) for automatic history matching. EnKF is a Monte Carlo method in which an ensemble of reservoir models is used. The correlation between reservoir response (e.g., water cut and rate) and reservoir variables (e.g., permeability and porosity) can be estimated from the ensemble. An estimate of uncertainty in future reservoir performance can also be obtained from the ensemble. The PUNQ-S3 reservoir model is used to test the method in this paper. It is a small (19×28×5) reservoir engineering model. One conclusion is that when applied to the PUNQ-S3 synthetic model, the EnKF technique gives satisfactory history-matching results while requiring less computation work than traditional methods. IntroductionThe process of adjusting the variables in a reservoir simulation model to honor observations of rates, pressures, saturations, and other variables at individual wells is called history matching. In many cases, general geological information also needs to be honored, such as the variance-covariance structure of the model parameters. Thus, to do history matching, one typically attempts to minimize the square of the mismatch between all measurements and computed values, and/or the square of the mismatch of the current model parameters and the prior model parameters. Although the process can now be largely automated, a large computational effort is still required, either in objective function evaluation (nongradient-based minimization method), or in gradient computation (gradient-based minimization method). If the gradient-based minimization methods are employed, the adjoint method may be required to compute the gradient of the objective function. The adjoint system is highly dependent on the source code of the reservoir simulator, however, and therefore is not flexible; that is, if we want to use a different simulator, development of an adjoint code requires considerable work. On the other hand, the increase in deployment of permanent sensors for monitoring pressure, temperature, resistivity, or flow rate has added impetus to the related problem of continuous model updating. Because the data output frequency in this case can be very high, to simultaneously use all recorded data to generate a reservoir flow model is impractical. Instead, it has become important to incorporate the data as soon as they are obtained so that the reservoir model is always up to date. Both the heavy computational burden and the high data-sampling frequency require a new kind of history-matching method.The Kalman filter has historically been the most widely applied method for assimilating new measurements to continuously update the estimate of state variables. Although Kalman filters have occasionally been applied to the problem of estimating values of petroleum model variables, 1,2 they are more suitable for the cases with small numbers of variables and a linear relationship between model and observations. Unfortunately, most problems in petroleum reservoir engineering are highly nonlin...

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe problem of reservoir characterization through automatic history matching has been extensively studied in recent years. Efficient applications have, however, required either an adjoint or a gradient simulator method to compute the gradient of the objective function or a sensitivity coefficient matrix for the minimization. Both computations are expensive when the number of model parameters or the number of observation data is large. The codes for gradient-based history matching methods are also complex and time-consuming to write. This paper reports the use of the Ensemble Kalman Filter (EnKF) for automatic history matching. EnKF is a Monte Carlo method, in which an ensemble of reservoir models is used. The correlation between reservoir response (e.g. watercut and rate) and reservoir variables (e.g. permeability and porosity) can be estimated from the ensemble. An estimate of uncertainty in future reservoir performance can also be obtained from the ensemble.The methodology of EnKF consists of a forecast step and an assimilation step. A finite-difference, 3-D, 3-phase black-oil reservoir simulator is used for stepping forward the reservoir states. However, unlike the traditional history matching, the source code of the reservoir simulator is not required, which allows this method to be used with any reservoir simulator. Moreover, this forward step is well suited for parallel computation since the time evolution of ensemble reservoir models are independent, hence the ensemble of reservoir models can be advanced in time simultaneously using multiple processors. Only the data assimilation step, i.e. the computation of Kalman filter, requires communication between processors.The assimilation of the data in EnKF is done sequentially rather than simultaneously as in traditional history matching.By so doing the reservoir models are always kept up-to-date, which is important and practical when the frequency of data is fairly high as, for example, the data from permanent sensors.The PUNQ-S3 reservoir model is used to test the method in this paper. It is a small-size (19 × 28 × 5) reservoir engineering model that was developed by a group of companies, institutes and universities in the European Union to compare methods for quantifying uncertainty assessment in history matching. One conclusion is that EnKF can sometimes provide satisfactory history matching results while requiring less computation work than traditional methods.

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