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.

From a system-theoretical point of view and for a given configuration of wells, there are only a limited number of degrees of freedom in the inputoutput dynamics of a reservoir system. This means that a large number of combinations of the state variables (pressure and saturation values) are not actually controllable and observable from the wells, and accordingly, they are not affecting the input-output behavior of the system. In an earlier publication, we therefore proposed a control-relevant upscaling methodology that uniformly coarsens the reservoir. Here, we present a control-relevant selective (i.e. non-uniform) coarsening (CRSC) method, in which the criterion for grid size adaptation is based on ranking the grid block contributions to the controllability and observability of the reservoir system. This multi-level CRSC method is attractive for use in iterative procedures such as computer-assisted flooding optimization for a given configuration of wells. In contrast to conventional flowbased coarsening techniques our method is independent of the specific flow rates or pressures imposed at the wells. Moreover the system-theoretical norms employed in our method provide tight upper bounds to the 'input-output energy' of the fine and coarse systems. These can be used as an a priori error-estimate of the performance of the coarse model. We applied our algorithm to two numerical examples and found that it can accurately reproduce results from the corresponding fine-scale simulations, while significantly speeding up the simulation.

No abstract