Summary
For large scale history matching problems, where it is not feasible to
compute individual sensitivity coefficients, the limited memory
Broyden-Fletcher-Goldfarb-Shanno (LBFGS) is an efficient optimization
algorithm, (Zhang and Reynolds, 2002; Zhang, 2002). However, computational
experiments reveal that application of the original implementation of LBFGS may
encounter the following problems: (i) converge to a model which gives an
unacceptable match of production data; (ii) generate a bad search direction
that either leads to false convergence or a restart with the steepest descent
direction which radically reduces the convergence rate; (iii) exhibit
overshooting and undershooting, i.e., converge to a vector of model parameters
which contains some abnormally high or low values of model parameters which are
physically unreasonable. Overshooting and undershooting can occur even though
all history matching problems are formulated in a Bayesian framework with a
prior model providing regularization.
We show that the rate of convergence and the robustness of the algorithm can
be significantly improved by: (1) a more robust line search algorithm motivated
by the theoretical result that the Wolfe conditions should be satisfied; (2) an
application of a data damping procedure at early iterations or (3) enforcing
constraints on the model parameters. Computational experiments also indicate
that (4) a simple rescaling of model parameters prior to application of the
optimization algorithm can improve the convergence properties of the algorithm
although the scaling procedure used can not be theoretically validated.
Introduction
Minimization of a smooth objective function is customarily done using a
gradient based optimization algorithm such as the Gauss- Newton (GN) method or
Levenberg-Marquardt (LM) algorithm. The standard implementations of these
algorithms (Tan and Kalogerakis, 1991; Wu et al., 1999; Li et al., 2003),
however, require the computation of all sensitivity coefficients in order to
formulate the Hessian matrix. We are interested in history matching problems
where the number of data to be matched ranges from a few hundred to several
thousand and the number of reservoir variables or model parameters to be
estimated or simulated ranges from a few hundred to a hundred thousand or more.
For the larger problems in this range, the computer resources required to
compute all sensitivity coefficients would prohibit the use of the standard
Gauss- Newton and Levenberg-Marquardt algorithms. Even for the smallest
problems in this range, computation of all sensitivity coefficients may not be
feasible as the resulting GN and LM algorithms may require the equivalent of
several hundred simulation runs. The relative computational efficiency of GN,
LM, nonlinear conjugate gradient and quasi-Newton methods have been discussed
in some detail by Zhang and Reynolds (2002) and Zhang (2002).
© 2006. Society of Petroleum Engineers
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History
- Original manuscript received:
6 June 2004
- Revised manuscript received:
26 July 2005
- Manuscript approved:
18 August 2005
- Version of record:
20 March 2006
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