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: (1) iteration to enforce
constraints and (2) ensuring 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.
© 2007. Society of Petroleum Engineers
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