Summary
History matching is an inverse problem in which an engineer calibrates key
geological/fluid flow parameters by fitting a simulator’s output to the real
reservoir production history. It has no unique solution because of insufficient
constraints. History-match solutions are obtained by searching for minima of an
objective function below a preselected threshold value. Experimental design and
response surface methodologies provide an efficient approach to build proxies
of objective functions (OF) for history matching. The search for minima can
then be easily performed on the proxies of OF as long as its accuracy is
acceptable.
In this paper, we first introduce a novel experimental design methodology
for semi-automatically selecting the sampling points, which are used to improve
the accuracy of constructed proxies of the nonlinear OF. This method is based
on derivatives of constructed proxies.
We propose an iterative procedure for history matching, applying this new
design methodology. To obtain the global optima, the proxies of an objective
function are initially constructed on the global parameter space. They are
iteratively improved until adequate accuracy is achieved. We locate subspaces
in the vicinity of the optima regions using a clustering technique to improve
the accuracy of the reconstructed OF in these subspaces.
We test this novel methodology and history-matching procedure with two
waterflooded reservoir models. One model is the Imperial College fault model
(Tavassoli et al. 2004). It contains a large bank of simulation runs. The other
is a modified version of SPE9 (Killough 1995) benchmark problem. We demonstrate
the efficiency of this newly developed history-matching technique.
Introduction
History matching (Eide et al. 1994; Landa and Güyagüler 2003) is an inverse
problem in which an engineer calibrates key geological/fluid flow parameters of
reservoirs by fitting a reservoir simulator’s output to the real reservoir
production history. It has no unique solution because of insufficient
constraints.
The traditional history matching is performed in a semi-empirical approach,
which is based on the engineer’s understanding of the field production
behavior. Usually, the model parameters are adjusted using a
one-factor-at-a-time approach. History matching can be very time consuming,
because many simulation runs may be required for obtaining good fitting
results.
Attempts have been made to automate the history-matching process by using
optimal control theory (Chen et al. 1974) and gradient techniques (Gomez et al.
2001). Also, design of experiment (DOE) and response surface methodologies
(Eide et al. 1994; Box and Wilson 1987; Montgomery 2001; Box and Hunter 1957;
Box and Wilson 1951; Damsleth et al. 1992; Egeland et al. 1992; Friedmann et
al. 2003) (RSM) were introduced in the late 1990s to guide automatic history
matching. The goal of these automatic methods is to achieve reasonably faster
history-matching techniques than the traditional method.
History matching is an optimization problem. The objective is to find the
best of all possible sets of geological/fluid flow parameters to fit the
production data of reservoirs. To assess the quality of the match, we define an
OF (Atallah 1999). For history-matching problems, an objective function is
usually defined as a distance (Landa and Güyagüler 2003) between a simulator’s
output and reservoir production data. History-matching solutions are obtained
by searching for minima of the objective function. Experimental design and
response surface methodologies provide an efficient approach to build up
hypersurfaces (Kecman 2001) of objective functions (i.e., proxies of objective
functions with a limited number of simulation runs for history matching). The
search for minima can then be easily performed on these proxies as long as
their accuracy is acceptable. The efficiency of this technique depends on
constructing adequately accurate objective functions.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
8 June 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
26 December 2006
- Manuscript approved:
12 January 2007
- Version of record:
20 December 2007
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