Summary
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.
Introduction
The 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 nonlinear and are characterized by
many variables, often two or more variables per simulator gridblock. Thus, the
traditional Kalman filters are not appropriate.
Application to nonlinear problems was at least partially solved by the
development of the extended Kalman filter. However, it did not solve the
critical problem with nonlinear unstable dynamics, in which it leads to a
linear instability in the error covariance evolution. The EnKF was introduced
to overcome some of the problems of the extended Kalman filter. 3 Since then,
the method has found widespread application in weather forcasting, 3--7
oceanography, 8 hydrology, 9 and petroleum engineering. 10,11
The EnKF has two major advantages for large-scale history-matching problems.
First, it does not depend on the specific reservoir simulator. It only requires
output from the simulator, such as pressure and phase saturation. Second, the
computational cost is fairly low. A relatively small ensemble might be
sufficient for most applications of EnKF. Although nongradient-based
minimization methods are also not dependent on the simulator source code, they
usually take thousands of simulation runs (objective function evaluations) to
obtain the global minimal point.
Naevdal et al. 11 applied the EnKF to the problem of updating 2D,
three-phase reservoir models by continuously adjusting both the permeability
field and the saturation and pressure fields at each assimilation step. In
their application, the porosity field is assumed to be known. One synthetic
example had 1,931 active gridblocks with 14 producers and four gas injectors.
Two of the producers obtained measurements of well pressure, oil rate, gas/oil
ratio, and water cut from the first day. Assimilation occurred at least once a
month as well as when new wells started to produce or when wells were shut in,
so in many respects it was quite similar to a traditional history-matching
problem. They found that the ability to predict future performance got steadily
better as more data were assimilated.
Ensemble Kalman Filter
The methodology consists of a forecast step (stepping forward in time) and
an assimilation step, in which variables describing the state of the system are
corrected to honor the observations.
The evolution of reservoir dynamic variables is dictated by reservoir-flow
equations and simulated using a commercial reservoir simulator in this
paper.
The following introduces the building blocks of the methodology.
© 2005. Society of Petroleum Engineers
View full textPDF
(
2,940 KB
)
History
- Original manuscript received:
7 June 2004
- Revised manuscript received:
1 March 2005
- Manuscript approved:
17 March 2005
- Version of record:
15 June 2005
View Cart
