  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 \cite{aes14}, 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. In this approach, the matrix and
the fracture interaction is modeled via local multiscale basis functions. We
developed the GMsFEM and applied for linear flows with horizontal or vertical
fracture orientations on a Cartesian fine grid. In this paper, we consider
arbitrary fracture orientations and use triangular fine grid and developed
GMsFEM for nonlinear flows. Moreover, we 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.
