Summary
It is well documented that history matching is a problem with possibly
nonunique solutions. In the past few years, several automated or semiautomated
history-matching algorithms have been proposed. Depending on the algorithm
used, it is possible that the final estimated reservoir-property distribution
that allows for a good history match may not be geologically realistic.
Therefore, there is a need to include other constraints to generate multiple,
geologically realistic history-matched realizations. These constraints might,
for example, include the variogram, a training image, the distribution of
net-to-gross, pore volume, or other geostatistical information about the
reservoir. This inclusion is particularly useful because it introduces
uncertainty information in the reservoir description when we have limited
history from existing wells in the field and intend to drill infill wells or
implement a secondary-recovery process.
The algorithm proposed in this paper uses multiresolution wavelet analysis
to integrate history data with the geostatistical information contained in the
variogram proposed for the reservoir. Wavelets allow the representation and
manipulation of property distributions at various resolutions at the same time.
Using wavelets, information from different sources such as production history
and seismic surveys (that would be at different resolutions) can be
incorporated directly at the appropriate resolution level. In the first step,
we fix the wavelet coefficients sensitive to the history-match data. This has
the effect of fixing the field history without fixing individual gridblock
properties. In the second step, the remaining free wavelet coefficients are
modified to integrate variogram information into the reservoir description.
Generating multiple realizations of only the second set of wavelets
coefficients results in multiple history-matched, variogram-constrained
descriptions of the reservoir. The computational investment is very modest
because the history match is done only once.
In a number of example cases, different areal Gaussian fields with varying
amounts of available production-history data were studied to test the
algorithm. It was found that the wavelet coefficients constraining the history
can be decoupled from those constraining the variogram. The implication of this
observation is that the history data and variogram can be integrated
sequentially into the reservoir model—that is, after the initial history match,
new information can be added to the model without disturbing the original match
to yield multiple history-matched and geostatistically constrained
realizations.
Introduction
Reservoir modeling is essential for forecasting the performance of a
reservoir, for reservoir management, for risk analysis, and for making key
economic decisions. The purpose of reservoir modeling is to develop a model of
the reservoir that closely resembles the actual reservoir based on available
information. This model then can be used to forecast future performance and
optimize reservoir-management decisions. The more accurate the reservoir model,
the better the predictions will be. Therein lies the importance of generating a
good reservoir model. History matching is but one step in this direction.
Merely achieving a good history match does not ensure sound predictions from
the reservoir model; it is therefore essential that all sources of information
about the reservoir be used appropriately to come up with a good model.
Early automated history-matching procedures were discussed by Jacquard and
Jain,1 adapted from variational analysis in electric networking. Since then,
there have been several developments of concepts and algorithms along similar
lines. In general, the objective is to determine the spatial distribution of a
set of gridded reservoir properties such as permeabilities and porosities,
given the response of the field in terms of fluid flow to an external impulse
such as drainage and injection of fluids, as well as geostatistical data.
Production history from existing wells is an important source of information
about the reservoir, in terms of the average permeabilities, spatial
distribution of permeabilities, net-to-gross, etc. Production history could be
in the form of the pressure or saturation distribution in the reservoir in
response to injection or production impulses. A good reservoir model must
therefore, when run through a flow simulator, give the same response to the
same impulse as the real reservoir. Many studies have shown favorable results
from integrating dynamic data into reservoir modeling using streamline
simulators (e.g., Datta-Gupta et al.2).
However, not only does history matching alone not ensure sound production
forecast, it also does not guarantee physical consistency and might produce
artifacts based on the algorithm used. The results thus obtained might give a
perfect history match, but if they are aphysical, use of the model will lead to
further error in prediction of future performance because the model may not be
close enough in a geological sense to the actual reservoir. This situation
arises because there may be a number of different solutions to the
history-matching problem. In other words, a number of different permeability
distributions may be found, all of which give the same response to a given
impulse. As such, we need to integrate geostatistical data that will constrain
the problem and make the model more realistic. Landa and Horne3 and Landa4
investigated the impact of different data on reservoir characterization and
uncertainty. Integration of static and dynamic data into reservoir models has
been attempted in the Bayesian framework5–7 and with gradual deformation.8
Multiresolution wavelet analysis forms the basis for efficient
representation of the field as well as a reduction in the number of parameters
to be estimated. As described in the following section, the gridded
reservoir-property distribution can be transformed linearly to give a unique
set of wavelet coefficients. It has been found9,10 that a specific subset of
these wavelet coefficients is sufficient to determine the response of the
reservoir to production. The conjecture is that the remaining set can be
modified subject to constraints based on geological, seismic, or other
subjective information about the spatial distribution of the permeabilities.
This study showed that the sets of wavelets constraining the history match and
those constraining geostatistical parameters (variograms in particular) can
indeed be decoupled and evaluated separately to yield a set of different
permeability distributions stochastically.
Most history-matching algorithms involve flow simulation at each iteration
while minimizing the objective function. The advantage of our new algorithm is
that instead of doing repeated history matches, it fixes a set of wavelet
coefficients that constrain the history, thereby fixing the history up to some
tolerance. The objective function endeavors to enforce a proposed variogram of
spatial distribution of the permeabilities. As such, the algorithm takes
orders-of-magnitude less time to yield permeability distributions that are
constrained by both the history and the variogram of the field.
© 2005. Society of Petroleum Engineers
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History
- Original manuscript received:
17 March 2004
- Revised manuscript received:
19 November 2004
- Manuscript approved:
22 November 2004
- Version of record:
15 February 2005
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