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.
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