Summary
The use of the ensemble Kalman filter (EnKF) is a promising approach for
data assimilation and assessment of uncertainties during reservoir
characterization and performance forecasting. It provides a relatively
straightforward approach to incorporating diverse data types, including
production and/or time-lapse seismic data. Unlike traditional sensitivity-based
history matching methods, the EnKF relies on a cross-covariance matrix computed
from an ensemble of reservoir models to relate reservoir properties to
production data. For practical field applications, we need to keep the ensemble
size small for computational efficiency. However, this leads to poor
approximations of the cross-covariance and, often, loss of geologic realism
through parameter overshoots, in particular by introducing localized patches of
low and high permeabilities. Because the EnKF estimates are "optimal"
only for Gaussian variables and linear dynamics, these difficulties are
compounded by the strong nonlinearity of the multiphase history matching
problems and for non-Gaussian prior models. Specifically, the updated parameter
distribution tends to become multi-Gaussian with loss of connectivities of
extreme values, such as high permeability channels and low permeability
barriers, which are of special significance during reservoir
characterization.
We propose a novel approach to overcome some of these limitations by
conditioning the cross-covariance matrix using information gleaned from
streamline trajectories. Our streamline-assisted EnKF is analogous to the
conventional assisted history matching, whereby the streamline trajectories are
used to identify gridblocks contributing to the production response of a
specific well. We then use these gridblocks only to compute the
cross-covariance matrix and eliminate the influence of unrelated or distant
observations and spurious correlations. We show that the streamline-assisted
EnKF is an efficient and robust approach for history matching and continuous
reservoir model updating. We illustrate the power and utility of our approach
using both synthetic and field applications.
Introduction
Proper characterization of the reservoir and the assessment of uncertainty
are crucial aspects of any optimal reservoir development plan and management
strategy. To achieve this goal, it is necessary to reconcile geological models
to the dynamic response of the reservoir through history matching. The topic of
history matching has been of great interest and an area of active research in
the oil industry (Datta-Gupta and King 2007; Emanuel and Milliken 1998; Oliver
et al. 2001). The past decade has seen some significant developments in
assisted and automatic history matching of high-resolution reservoir models and
associated uncertainty quantification. Many of these techniques involve
computation of sensitivities that relate changes in production response at a
well to a change in reservoir parameters. Techniques of automatic history
matching that typically do not use parameter sensitivities or gradient of the
misfit function are stochastic algorithms such as Markov Chain Monte Carlo
(MCMC), simulated annealing and genetic algorithms (Ma et al. 2008; Sen et al.
2005). A relatively recent and promising addition to this class of techniques
is the use of ensemble Kalman Filters (EnKF) for data assimilation (Gu and
Oliver 2005, 2006; Naevdal et al. 2005; Gao et al. 2006; Skjervheim et al.
2007; Dong et al. 2006). It is a Monte-Carlo approach that works with an
ensemble of reservoir models. Specifically, the method utilizes
cross-covariances between measurements and model parameters computed directly
from the ensemble members to sequentially update the reservoir models.
A major advantage of the EnKF is that it can be readily linked to any
existing reservoir simulator. The ability to assimilate diverse data types and
the ease of implementation have resulted in considerable interest in the
approach. Moreover, EnKF uses a sequential updating technique; that is, the
reservoir data is assimilated as and when it becomes available. The EnKF can
assimilate the latest production data without re-running the simulator from the
initial conditions. These characteristics make it particularly well-suited for
continuous model updating. The increased application of downhole monitors,
intelligent well systems, and permanent sensors to continuously record
pressure, well rates, and temperature has provided a further boost to the
sequential model updating through EnKF.
In spite of all its favorable properties, the current implementation of EnKF
approach comes with its own share of challenges. A key requirement in history
matching is that the final model should honor the available geological
information and retain geologic realism. It has been shown that the EnKF works
well when the prior distribution of parameters is Gaussian; however, the
estimates are suboptimal for non-Gaussian distributions. Over a sequence of
many updates, multimodal permeability distributions tend to transform to
Gaussian distribution. During geologic model updating, this can lead to a loss
of structure and connectivity of the extremes in the permeability field. This
has serious implications in the fluid flow because of the influence of
high-permeability channels and low-permeability barriers. Although there are
some variants of the Kalman filter that work with non-Gaussian distributions,
such as the Gaussian summation approximation, the implementation on an ensemble
framework tend to be very expensive (Anderson and Moore 1979).
In the past few years, we have seen several applications of the EnKF for
field-scale history matching, including some recent papers that attempt to deal
with some of the challenges pertaining to its use (Gu and Oliver 2005, 2006;
Naevdal et al. 2005; Gao et al. 2006; Skjervheim et al. 2007; Dong et al.
2006). In particular, localized overshooting of permeabilities has been
reported, resulting in loss of geologic continuity. This is aggravated by the
strong non-linearity inherent in multiphase flow simulations.
Another common difficulty experienced when using the EnKF is filter
divergence. The effect of filter divergence is such that the distribution
produced by the filter drifts away from the truth. Filter divergence normally
occurs because the prior probability distribution becomes too narrow (loss of
variance) and the observations have progressively less impact on the model
updates.
One common approach to deal with filter divergence is to add some (white)
noise to the prior ensemble to "inflate" its distribution and enhance
the impact of new observations. Other problems and limitations of the EnKF,
particularly for nonlinear problems and non-Gaussian parameter distributions,
can be partly controlled using a large ensemble. However, for practical field
applications, the ensemble size needs to be kept relatively small for
computational efficiency.
This paper describes an approach to address many of the currently reported
difficulties in the use of the EnKF applied to reservoir history matching. The
unique feature of our proposed approach is that the final models that
constitute the ensemble tend to retain the geological information that went
into building them initially. Over a sequence of many updates, our approach
tends to preserve the shape of the initial permeability distribution and
consequently retains key geological features. Our approach greatly decreases
the severity of the overshooting problem reported in earlier implementations of
the EnKF. Moreover, it allows the use of smaller ensemble size, while providing
results comparable or better than the standard EnKF.
The paper is organized as follows. First, we briefly review the major steps
of the EnKF and the additional streamline-based conditioning of the
cross-covariance proposed here. We also illustrate these steps using a
synthetic example. Next, we discuss the underlying mathematical formulation in
detail. We then demonstrate the power and practical utility of the approach
using the benchmark PUNQ-S3 synthetic example (Gu and Oliver 2005) and a field
example. Finally, an analysis of the scalability and speed-up factor for the
parallel implementation of our code is given.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
21 August 2006
- Meeting paper published:
5 December 2006
- Revised manuscript received:
8 July 2008
- Manuscript approved:
22 July 2008
- Version of record:
29 December 2008
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