Summary
The ensemble Kalman filter (EnKF) has gained increased popularity for
history matching and continuous reservoir-model updating. It is a sequential
Monte Carlo approach that works with an ensemble of reservoir models.
Specifically, the method uses cross covariance between measurements and model
parameters estimated from the ensemble. For practical field applications, the
ensemble size needs to be kept small for computational efficiency. However,
this leads to poor approximations of the cross covariance and can cause loss of
geologic realism from unrealistic model updates outside the region of the data
influence and/or loss of variance leading to ensemble collapse. A common
approach to remedy the situation is to limit the influence of the data through
covariance localization.
In this paper, we show that for three-phase-flow conditions, the region of
covariance localization strongly depends on the underlying flow dynamics as
well as on the particular data type that is being assimilated in terms of water
cut or gas/oil ratio (GOR). This makes the traditional distance-based
localizations suboptimal and, often, ineffective. Instead, we propose the use
of water- and gas-phase streamlines as a means for covariance localization for
water-cut- and GOR-data assimilation. The phase streamlines can be computed on
the basis of individual-phase velocities which are readily available after flow
simulation. Unlike the total streamlines, phase streamlines can be
discontinuous. We show that the discontinuities in water-phase and gas-phase
streamlines naturally define the region of influence for water-cut and GOR data
and provide a flow-relevant covariance localization during EnKF updating.
We first demonstrate the validity of the proposed localization approach
using a waterflood example in a quarter-five-spot pattern. Specifically, we
compare the phase streamline trajectories with cross-covariance maps computed
using an ensemble size of 2,000 for both water-cut and GOR data. The results
show a close correspondence between the time evolution of phase streamlines and
the cross-covariance maps of water-cut and GOR data. A small-size industrial
reservoir engineering production forecasting with uncertainty quantification
(the PUNQ-S3) (Carter 2007) model application shows that our proposed
localization outperforms a distance-based localization method. The updated
models show improved forecasts while preserving geological realism.
© 2012. Society of Petroleum Engineers
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History
- Original manuscript received:
24 August 2011
- Meeting paper published:
8 May 2011
- Revised manuscript received:
2 December 2011
- Manuscript approved:
17 January 2012
- Published online:
24 May 2012
- Version of record:
12 June 2012
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