In this paper, we develop a multiscale model reduction technique that describes shale gas transport in fractured media. Due to the pore-scale heterogeneities and processes, we use upscaled models to describe the matrix. We follow our previous work (Akkutlu et al. Transp. Porous Media 107(1), 235–260, 2015), where we derived an upscaled model in the form of generalized nonlinear diffusion model to describe the effects of kerogen. To model the interaction between the matrix and the fractures, we use Generalized Multiscale Finite Element Method (Efendiev et al. J. Comput. Phys. 251, 116–135, 2013, 2015). In this approach, the matrix and the fracture interaction is modeled via local multiscale basis functions. In Efendiev et al. (2015), we developed the GMsFEM and applied for linear flows with horizontal or vertical fracture orientations aligned with a Cartesian fine grid. The approach in Efendiev et al. (2015) does not allow handling arbitrary fracture distributions. In this paper, we (1) consider arbitrary fracture distributions on an unstructured grid; (2) develop GMsFEM for nonlinear flows; and (3) develop online basis function strategies to adaptively improve the convergence. The number of multiscale basis functions in each coarse region represents the degrees of freedom needed to achieve a certain error threshold. Our approach is adaptive in a sense that the multiscale basis functions can be added in the regions of interest. Numerical results for two-dimensional problem are presented to demonstrate the efficiency of proposed approach. © 2016 Springer International Publishing Switzerland
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: We are grateful to Tat Leung for helpful discussions and suggestions regarding to online basis constructions. MV's work is partially supported by the grant of the President of the Russian Federation MK-9613.2016.1 and RFBR (project N 15-31-20856). YE would like to thank the partial support from the DOE, Army, the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165, and National Priorities Research Program grant NPRP grant 7-1482-1278 from the Qatar National Research Fund (a member of The Qatar Foundation).