Summary
Smart fields can provide enhanced oil recovery through the combined use of
optimization and data assimilation. In this paper, we focus on the dynamic
optimization of injection and production rates during waterflooding. In
particular, we use optimal control theory in order to find an optimal well
management strategy over the life of the reservoir that maximizes an objective
function (e.g., recovery or net present value). Optimal control requires the
determination of a potentially large number of (groups of) well rates for a
potentially large number of time periods. However, the optimal number of well
groups and time steps is not known a priori. Moreover, taking these
numbers too large can slow down the optimization process and increase the
chance of achieving a suboptimal solution. We investigate the use of multiscale
regularization methods to achieve grouping of the control settings of the wells
in both space and time. Starting out with a very coarse grouping, the
resolution is subsequently refined during the optimization. The regularization
is adaptive in that the multiscale parameterization is chosen based on the
gradients of the objective function. Results for the numerical examples studied
indicate that the regularization may lead to significantly simpler optimum
strategies, while resulting in a better or similar cumulative oil
production.
Introduction
We consider the secondary recovery phase of a heterogeneous oil reservoir,
where water is injected into the reservoir for pressure maintenance and sweep
improvement. In a smart field scenario, we consider injectors and producers
with both single and multiple completions. The flow rates of the different well
completions can be adjusted individually. In the following, an individual well
completion will be referred to as "well segment." This implies that in
case of conventional single-completion wells the term "well segment" is
therefore equivalent to "well." Ideally, the injected water will
displace the remaining oil in the reservoir on its way from the injection wells
to the production wells. Rock heterogeneities will, however, influence the path
of the injected water. The water will mainly flow in the high-permeability
channels, which causes only part of the oil to be produced. Recently, smart
field concepts have been proposed as a means to improve control over the
waterfront through detailed adjustments of the injection and production rates
in time using a combination of model-based flooding optimization and model
updating (Brouwer et al. 2004; Sarma et al. 2005b). For the optimization part,
these "closed-loop" reservoir management strategies rely on optimal
control theory, which has been proposed before as a flooding optimization
method by various authors (Asheim 1988; Virnovski 1991; Sudaryanto 1998;
Brouwer et al. 2004; Sarma et al. 2005a). However, optimization by means of
optimal control theory is computationally expensive, and detailed management of
every individual well segment of a smart field at every moment in time is
economically and technically demanding. Moreover, there may not be enough
information in the system to determine the optimal production strategy
uniquely. Hence, we seek to develop management strategies with a restricted
number of degrees of freedom, which at the same time maintain the advantages of
the smart field technology.
In this paper, multiscale estimation techniques are utilized to attempt to
find the optimal well management level. These are hierarchical regularization
methods where the number of degrees of freedom in the estimation is gradually
increased as the optimization proceeds. Multiscale methods were first applied
for solving partial differential equations to speed up convergence (Brandt
1977; Briggs 1987). Later, through the development of wavelets, multiscale
approaches have also been widely used within inverse problems (Emsellem and de
Marsily 1971; Chavent and Liu 1989; Liu 1993; Yoon et al. 2001).
The outline of the paper is as follows: First, the theory behind the
solution of the problem in terms of optimal control and gradient-based
optimization is presented. Thereafter we present methods to regularize the
optimization problem in terms of multiscale reparameterization of the control
variable. Finally, the performance of the proposed regularization strategies is
illustrated through a line of numerical examples before we summarize and
conclude.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
25 October 2006
- Meeting paper published:
11 April 2006
- Revised manuscript received:
19 June 2007
- Manuscript approved:
18 December 2007
- Version of record:
25 June 2008
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