Operator Based Multiscale Method for Compressible Flow

Accurate simulation of subsurface flow with detailed geologic description is of great academic and industrial interest. Fully fine-scale simulation is usually too expensive. The multiscale method is developed to capture fine-scale information without solving fine-scale equations. It is more efficient than fine-scale simulation methods and more accurate than traditional upscaling techniques. Previous existing multiscale methods deal with the incompressible flow problems only. However, compressibility will be very significant if one of the fluid phases is gas. Gas has a large compressibility, and its compressibility is usually a strong function of pressure. Therefore, there can be a significant spatial variation of compressibility in the reservoir, and this is a challenge for multiscale modeling. Motivated to construct a general multiscale framework that can deal with complicated physics, we develope an Operator Based Multiscale Method (OBMM). In this method, we first construct two multiscale operators — prolongation and restriction. Using the two operators, we construct the coarse-scale equations from fine-scale equations through simple algebraic operations. Then, we solve for the coarse-scale variables and reconstruct the fine-scale solution as in previous methods. The algorithm is succinct, and it is a general algebraic framework. As an example, compressibility can be naturally included into this framework. Numerical examples show the accuracy and efficiency of this method.