Upscaling of Large Reservoir Systems by Using a Control-relevant Approach

In reservoir simulations, we use an ensemble of fine-scale geological models that are upscaled to coarser representations for flow simulations. The primary objective of the upscaling process is to meet the computational limits of the simulator. However, we argue that from a system-theoretic point of view, a more fundamental underlying reason for upscaling is that the complexity level of a model should be adjusted to the amount of available information from measurements and the extent of control (input) exercised by adjusting the well parameters. That is because for a given configuration of wells, a large number of combinations of state variables (pressure and saturation values in the gridblocks) are not actually controllable and observable and accordingly they are not affecting the input-output behavior of the model. Therefore, we propose a “control-relevant-upscaling” (CRU) approach that determines equivalent coarse-scale permeabilities based on the actual system’s input-output behavior. The coarse-scale parameters are obtained 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 by using Hankel- or energy norms, in which we use Hankel singular values and Markov parameters as a measure of the combined controllability and observability, and response of the system, respectively. This work focuses on single-phase flow upscaling, where we develop a CRU algorithm for reservoir systems. Moreover, we address the potential benefits of using proper orthogonal decomposition (POD) in combination with our CRU method to obtain a reduced-order CRU algorithm that accelerate the upscaling procedure. In the cases considered, the CRU algorithm shows superior input-output behavior as compared to upscaling algorithms commonly used in reservoir simulators.