Control-Relevant Upscaling

Abstract

Summary 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 behaviors of the fine- and coarse-scale models. This distance is measured with the aid of the Hankel or energy norms, in which we use Hankel singular values as a measure of the combined controllability and observability and Markov parameters as a measure of the response of the system, respectively. The method is particularly attractive to scale up simulation models in flooding-optimization or history-matching studies for a given configuration of wells. An advantage of our upscaling method is that it 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, either of the full or of the upscaled model. It also does not depend on a particular control strategy but instead uses the dynamical system equations directly. Its dependency on well locations, however, implies that it should be (partially) repeated when those locations 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.

A system-theoretical approach to selective grid coarsening of reservoir models

“…Therefore, the complexity level of a model should be adjusted to the amount of available information and the extent of control that is possible in the reservoir system. In an earlier publication, we therefore proposed a control-relevant upscaling (CRU) method that uniformly coarsens the reservoir model based on the relevant level of information and control (see [37]). Note that we use the terms upscaling and coarsening interchangeably.…”

“…Its dependency on well locations, however, implies that it should be (partially) repeated when those locations are changed. Moreover, the formulation as presented in Vakili-Ghahani and Jansen [37] is restricted to fine-scale models with a maximum of around 10 5 grid blocks because of current limits on the computation of the underlying system norms. In VakiliGhahani et al [35] we proposed to overcome this computational limit by combining CRU with model-order reduction techniques.…”

“…[2,33]. For recent applications of system theory to reservoir modeling see [6,14,19,24,25,34,37,38,43].…”

“…The HSVs can give a measure of the combined controllability and observability of the balanced states of the system. In [35,37] we used this analysis to develop a uniform CRU method that indirectly focuses on the most controllable and observable states of the system. There we found the coarse grid block parameters by minimizing the difference between the fine and the coarse reservoir models in terms of a system norm that characterizes the input-output behavior for a given configuration of wells.…”

“…In most reservoir models, the magnitude of HSVs are rapidly decreasing, indicating that a large number of the states are weakly controllable/observable. Hence, they weakly influence the input-output behavior of the system [34,39,43]. Considering this effect, we can rewrite Eq.…”