Summary
The Bayesian framework allows one to integrate production and static data
into an a posteriori probability density function (pdf) for reservoir variables
(model parameters). The problem of generating realizations of the reservoir
variables for the assessment of uncertainty in reservoir description or
predicted reservoir performance then becomes a problem of sampling this a
posteriori pdf to obtain a suite of realizations. Generation of a realization
by the randomized-maximum-likelihood method requires the minimization of an
objective function that includes production-data misfit terms and a model
misfit term that arises from a prior model constructed from static data.
Minimization of this objective function with an optimization algorithm is
equivalent to the automatic history matching of production data, with a prior
model constructed from static data providing regularization. Because of the
computational cost of computing sensitivity coefficients and the need to solve
matrix problems involving the covariance matrix for the prior model, this
approach has not been applied to problems in which the number of data and the
number of reservoir-model parameters are both large and the forward problem is
solved by a conventional finite-difference simulator.
In this work, we illustrate that computational efficiency problems can be
overcome by using a scaled limited-memory Broyden-Fletcher-Goldfarb-Shanno
(LBFGS) algorithm to minimize the objective function and by using approximate
computational stencils to approximate the multiplication of a vector by the
prior covariance matrix or its inverse. Implementation of the LBFGS method
requires only the gradient of the objective function, which can be obtained
from a single solution of the adjoint problem; individual sensitivity
coefficients are not needed. We apply the overall process to two examples. The
first is a true field example in which a realization of log permeabilities at
26,019 gridblocks is generated by the automatic history matching of pressure
data, and the second is a pseudofield example that provides a very rough
approximation to a North Sea reservoir in which a realization of log
permeabilities at 9,750 gridblocks is computed by the automatic history
matching of gas/oil ratio (GOR) and pressure data.
Introduction
The Bayes theorem provides a general framework for updating a pdf as new
data or information on the model becomes available. The Bayesian setting offers
a distinct advantage. If one can generate a suite of realizations that
represent a correct sampling of the a posteriori pdf, then the suite of samples
provides an assessment of the uncertainty in reservoir variables. Moreover, by
predicting future reservoir performance under proposed operating conditions for
each realization, one can characterize the uncertainty in future performance
predictions by constructing statistics for the set of outcomes. Liu and Oliver
have recently presented a comparison of methods for sampling the a posteriori
pdf. Their results indicate that the randomized-maximum-likelihood method is
adequate for evaluating uncertainty with a relatively limited number of
samples. In this work, we consider the case in which a prior geostatistical
model constructed from static data is available and is represented by a
multivariate Gaussian pdf. Then, the a posteriori pdf conditional to
production data is such that calculation of the maximum a posteriori estimate
or generation of a realization by the randomized-maximum-likelihood method is
equivalent to the minimization of an appropriate objective function.
History-matching problems of interest to us involve a few thousand to tens
of thousands of reservoir variables and a few hundred to a few thousand
production data. Thus, an optimization algorithm suitable for large-scale
problems is needed. Our belief is that nongradient-based algorithms such as
simulated annealing and the genetic algorithm are not competitive with
gradient-based algorithms in terms of computational efficiency. Classical
gradient-based algorithms such as the Gauss-Newton and Levenberg-Marquardt
typically converge fairly quickly and have been applied successfully to
automatic history matching for both single-phase- and multiphase-flow problems.
No multiphase-flow example considered in these papers involved more than 1,500
reservoir variables. For single-phase-flow problems, He et al. and Reynolds et
al. have generated realizations of models involving up to 12,500 reservoir
variables by automatic history matching of pressure data. However, they used a
procedure based on their generalization of the method of Carter et al. to
calculate sensitivity coefficients; this method assumes that the
partial-differential equation solved by reservoir simulation is linear and does
not apply for multiphase-flow problems.
© 2005. Society of Petroleum Engineers
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History
- Original manuscript received:
31 March 2004
- Revised manuscript received:
18 January 2005
- Manuscript approved:
27 January 2005
- Version of record:
15 June 2005
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