Control-Relevant Upscaling

Abstract

Abstract The conventional reason for upscaling in reservoir simulation is the computational limit of the simulator. However, we argue that, from a system-theoretical point of view, a more fundamental reason is that there is only a limited amount of information (output) that can be observed from production data, while there is also a limited amount of control (input) that can be exercised by adjusting the well parameters; in other words, the input-output behavior is usually of much lower dynamical order than the number of grid blocks in the model. Therefore, we propose an upscaling approach to find a coarse model that optimally describes the input-output behavior of a reservoir system. In this control-relevant method, the coarse-scale model parameters are calculated as the solution of an optimization problem that minimizes the distance between the input-output behavior of the fine- and coarse-scale models. This distance is measured with the aid of the Hankel- or energy norms, where Hankel singular values are used as a measure of the combined controllability and observability of the system. The method is particularly attractive to scale up simulation models in flooding optimization and/or history matching studies for a given configuration of wells. An advantage of our upscaling method is that it intrinsically relies most heavily on those parameter values that directly influence the input-output behavior. It is a global method, in the sense that it relies on the system properties of the entire reservoir. It does not, however, require any forward simulation, neither of the full nor of the upscaled model. it also does not depends on a particular control strategy, but instead uses the dynamical system equations directly. A drawback, however, is its dependency on well positions, which implies that it should be (partially) repeated when those positions are changed. We tested the method on several examples and for nearly all cases obtained coarse scale models with a superior input-output behavior compared to common upscaling algorithms.