  In this paper, we propose a rigorous and accurate non-local (in the
oversampled region) upscaling framework based on some recently developed
multiscale methods [10]. Our proposed method consists of identifying
multi-continua parameters via local basis functions and constructing non-local
(in the oversampled region) transfer and effective properties. To achieve this,
we significantly modify our recent work proposed within Generalized Multiscale
Finite Element Method (GMsFEM) in [10] and derive appropriate local problems in
oversampled regions once we identify important modes representing each
continua. We use piecewise constant functions in each fracture network and in
the matrix to write an upscaled equation. Thus, the resulting upscaled equation
is of minimal size and the unknowns are average pressures in the fractures and
the matrix. We note that the use of non-local upscaled model for porous media
flows is not new, e.g., in [14], the authors derive non-local approach. Our
main contribution is identifying appropriate local problems together with local
spectral modes to represent each continua. The model problem for fractures
assumes that one can identify fracture networks. The resulting non-local
equation (restricted to the oversampling region, which is several times larger
compared to the target coarse block) has the same form as \cite{Hamdi_Nonlocal}
with much smaller local regions. We present numerical results, which show that
the proposed approach can provide good accuracy.
