Introduction and Background
There has been great progress in data assimilation within atmospheric
and oceanographic sciences during the last couple of decades. In data
assimilation, one aims at merging the information from observations into a
numerical model, typically of a geophysical system. A typical example where
data assimilation is needed is in weather forecasting. Here, the atmospheric
models must take into account the most recent observations of variables such as
temperature and atmospheric pressure for better forecasting of the weather in
the next time period. A major challenge for these models is that they contain
very large numbers of variables.
The progress in data assimilation is because of both increased computational
power and the introduction of techniques that are capable of handling large
amounts of data and more severe nonlinearities. The aim of this paper is to
focus on one of these techniques, the ensemble Kalman filter (EnKF). The EnKF
has been introduced to petroleum science recently (Lorentzen et al. 2001a) and,
in particular, has attracted attention as a promising method for solving the
history matching problem. The literature available on the EnKF is now rather
overwhelming. We hope that this review will help researchers (and students)
working on adapting the EnKF to petroleum applications to find valuable
references and ideas, although the number of papers discussing the EnKF is too
large to give a complete review. For practitioners, we have cited critical EnKF
papers from weather and oceanography. We have also tried to review most of the
papers dealing with the EnKF and updating of reservoir models available to the
authors by the beginning of 2008.
The EnKF is based on the simpler Kalman filter (Kalman 1960). We will start
by introducing the Kalman filter. The Kalman filter is an efficient recursive
filter that estimates the state of a linear dynamical system from a series of
noisy measurements. The Kalman filter is based on a model equation, where the
current state of the system is associated with an uncertainty (expressed by a
covariance matrix) and an observation equation that relates a linear
combination of the states to measurements. The measurements are also associated
with uncertainty. The model equations are used to compute a forward step (Eqs.
1 and 2) where the state variables are computed forward in time with the
current estimate of the state as initial condition. The observation equations
are used in the analysis step (Eqs. 3 through 5) where the estimated value of
the state and its uncertainty are corrected to take into account the most
recent measurements See, e.g., Cohn (1997), Maybeck (1979), or Stengel (1994)
for an introduction to the Kalman filter.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
27 March 2008
- Revised manuscript received:
21 December 2008
- Manuscript approved:
1 January 2009
- Published online:
28 August 2009
- Version of record:
28 September 2009
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