Summary
The objective of this paper is to compare the performance of the ensemble
Kalman filter (EnKF) to the performance of a gradient-based minimization method
for the problem of estimation of facies boundaries in history matching. The
EnKF is a Monte Carlo method for data assimilation that uses an ensemble of
reservoir models to represent and update the covariance of variables. In
several published studies, it outperforms traditional history-matching
algorithms in adaptability and efficiency.
Because of the approximate nature of the EnKF, the realizations from one
ensemble tend to underestimate uncertainty, especially for problems that are
highly nonlinear. In this paper, the distributions of reservoir-model
realizations from 20 independent ensembles are compared with the distributions
from 20 randomized-maximum-likelihood (RML) realizations for a 2D waterflood
model with one injector and four producers. RML is a gradient-based sampling
method that generates one reservoir realization in each minimization of the
objective function. It is an approximate sampling method, but its sampling
properties are similar to the Markov-chain Monte Carlo (McMC) method on highly
nonlinear problems and are relatively more efficient than McMC.
Despite the nonlinear relationship between the data (such as production
rates and facies observations) and the model variables, the EnKF was effective
at history matching the production data. We find that the computational effort
to generate 20 independent realizations was similar for the two methods,
although the complexity of the code is substantially less for the EnKF.
Introduction
Several questions regarding the use of the EnKF for history matching are
addressed in this paper. The most important is a comparison of the efficiency
with a gradient-based method for a history-matching problem with known facies
properties but unknown boundary locations. Secondly, the EnKF and a
gradient-based method are unlikely to give identical estimates of model
variables, so it is also important to know if one method generates better
realizations. Finally, because there is often a desire to use the
history-matched realizations to quantify uncertainty, it is important to
determine if one of the methods is more efficient at generating independent
realizations.
Gradient-based history matching can be performed in several ways (e.g.,
assimilating data in batch or sequentially); a variety of minimization
algorithms can be used (e.g., conjugate gradient or quasi-Newton); and several
different methods for computing the gradient are available (e.g., adjoint or
sensitivity equations). In this paper, we use what we believe is the most
efficient of the traditional gradient-based methods: an adjoint method to
compute the gradient of the squared data mismatch and the limited-memory
Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method to compute the direction of the
change. The remaining choice is whether to incorporate all data at once or
sequentially. Simultaneous, or batch, inversion of all data is clearly a
well-established history-matching procedure. Although data from wells or
sensors may arrive nearly continuously, the practice of updating reservoir
models as the data arrive is not common. There are several reasons that make
sequential assimilation of data difficult for large, nonlinear models: (1) the
covariance for all model variables must be updated as new data are assimilated,
but the covariance matrix is very large; (2) the covariance may not be a good
measure of uncertainty for nonlinear problems; and (3) the sensitivity of a
datum to changes in values of model variables is expensive to compute.
Bayesian updating in general is described by Woodbury. Modifying a method
described by Tarantola, Oliver evaluated the possibility of using a sequential
assimilation approach for transient flow in porous media. He found that the
results from sequential assimilation could be almost as good as those from
batch assimilation if the order of the data was carefully selected. The problem
was quite small, however, and an extension to large models was impractical.
Although a sequential method has the advantage of generating a sequence of
history-matched models that may all be useful at the time they are generated,
our comparisons of efficiency will be based primarily on the effort required to
assimilate all the data. If the intermediate predictions are needed (as they
would be for control of a reservoir), the comparison provided here will
underestimate the value of the sequential assimilation.
A secondary objective of history matching is often to assess the uncertainty
in the predictions of future reservoir performance or in the estimates of
reservoir properties such as permeability, porosity, or saturation. In general,
uncertainty is estimated from an examination of a moderate number of
conditional simulations of the prediction or properties. Unless the
realizations are generated fairly carefully and the sample is sufficiently
large, however, the estimate of uncertainty could be quite poor. Two large
comparative studies of the ability of Monte Carlo methods to quantify
uncertainty in history matching have been carried out, one in groundwater and
one in petroleum. Neither was conclusive, partly because of the small sample
size. Liu and Oliver used a smaller reservoir model (fewer variables), but a
much larger sample size. They found that the method that minimizes an objective
function containing a model mismatch part and a data mismatch part, with noise
added to observations, created realizations that were distributed nearly the
same as realizations from McMC.
The EnKF is a Monte Carlo method for updating reservoir models. It solves
several problems with the application of the Kalman filter to large nonlinear
problems. It has been applied to reservoir flow problems with generally good
results. There has been no examination, however, of the distribution of the
members of a single ensemble. The adequacy of the uncertainty estimate is
completely unknown. In the first paper on the EnKF, Evensen described how the
evolution of the probability density function for the model variables can be
approximated by the motion of “particles” or ensemble members in phase space.
Any desired statistical quantities can be estimated from the ensemble of
points. When the size of the ensemble is relatively small, however, the
approximation of the covariance from the ensemble almost certainly contains
substantial errors. Houtekamer and Mitchell noted the tendency for a reduction
in variance caused by “inbreeding.” When the ensemble estimate is used in a
Kalman filter, van Leeuwen explained how nonlinearity in the covariance update
relation causes growth in the error as additional data are assimilated. In this
paper, the comparison is made using history matching on a truncated
pluri-gaussian model for geologic facies. It provides a difficult
history-matching problem with significant nonlinearities that make both the
EnKF and the LBFGS method difficult to apply.
© 2005. Society of Petroleum Engineers
View full textPDF
(
967 KB
)
History
- Original manuscript received:
7 December 2004
- Revised manuscript received:
13 August 2005
- Manuscript approved:
30 August 2005
- Version of record:
15 December 2005
View Cart
